Algebra Examples

$[\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}]$

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A + (- λ I_{4}))$

Substitute the known values in the formula.

$p (λ) = determinant ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Subtract the eigenvalue times the identity matrix from the original matrix.

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Multiply $- λ$ by each element of the matrix.

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify each element of the matrix $[\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$ .

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Simplify element $0, 0$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $0, 1$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $0, 2$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $0, 3$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 0$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 1$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 2$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 3$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 0$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 1$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 2$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 3$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $3, 0$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $3, 1$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $3, 2$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Simplify element $3, 3$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Combine the similar matrices with each others.

Simplify each element of the matrix $[\begin{array}{c} - λ - 2 & 0 + 1 & 0 - 2 & 0 - 4 \\ 0 + 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}]$ .

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Combine the same size matrices $[\begin{array}{c} - 2 & 1 & - 2 & - 4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & - 2 & - 4 \\ 3 & - 4 & 5 & 10 \end{array}]$ and $[\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}]$ by adding the corresponding elements of each.

$p (λ) = ([\begin{array}{c} - λ - 2 & 0 + 1 & 0 - 2 & 0 - 4 \\ 0 + 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $0, 1$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & 0 - 2 & 0 - 4 \\ 0 + 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $0, 2$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & 0 - 4 \\ 0 + 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $0, 3$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 0 + 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $1, 0$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 0 + 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $1, 2$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 0 + 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $1, 3$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 0 + 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $2, 0$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 0 + 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $2, 1$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & 0 - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $2, 3$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & - 4 \\ 0 + 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $3, 0$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & - 4 \\ 3 & 0 - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $3, 1$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & - 4 \\ 3 & - 4 & 0 + 5 & - λ + 10 \end{array}])$

Simplify element $3, 2$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & - 4 \\ 3 & - 4 & 5 & - λ + 10 \end{array}])$

The determinant of $[\begin{array}{c} - λ - 2 & 1 & - 2 & - 4 \\ 12 & - λ + 1 & 4 & 9 \\ 6 & 5 & - λ - 2 & - 4 \\ 3 & - 4 & 5 & - λ + 10 \end{array}]$ is $λ^{4} - 7 λ^{3} + 18 λ^{2} - 20 λ + 8$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = (- λ - 2) | \begin{array}{c} - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $(- λ - 2) | \begin{array}{c} - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} |$ is $λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = (- λ - 2) ((- λ + 1) | \begin{array}{c} - λ - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $(- λ + 1) | \begin{array}{c} - λ - 2 & - 4 \\ 5 & - λ + 10 \end{array} |$ is $- λ^{3} + 9 λ^{2} - 8 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (- λ - 2) ((- λ + 1) ((- λ - 2) (- λ + 10) - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify terms.

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Simplify each term.

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Expand $(- λ - 2) (- λ + 10)$ using the FOIL Method.

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Apply the distributive property.

$p (λ) = (- λ - 2) ((- λ + 1) (- λ (- λ + 10) - 2 (- λ + 10) - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) ((- λ + 1) (- λ (- λ) - λ \cdot 10 - 2 (- λ + 10) - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) ((- λ + 1) (- λ (- λ) - λ \cdot 10 + (- 2 (- λ) - 2 \cdot 10) - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Remove parentheses.

$p (λ) = (- λ - 2) ((- λ + 1) (- λ (- λ) - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify and combine like terms.

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Simplify each term.

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Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = (- λ - 2) ((- λ + 1) (- λ λ \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (- λ - 2) ((- λ + 1) (- λ^{1 + 1} \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $1$ and $1$ .

$p (λ) = (- λ - 2) ((- λ + 1) (- λ^{2} \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- λ^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = (- λ - 2) ((- λ + 1) (1 λ^{2} - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ^{2}$ by $1$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $10$ by $- 1$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 10 λ - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $- 2$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 10 λ + 2 λ - 2 \cdot 10 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $10$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 10 λ + 2 λ - 20 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 10 λ$ and $2 λ$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 8 λ - 20 - 5 \cdot - 4) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $- 4$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 8 λ - 20 + 20) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by adding numbers.

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Add $- 20$ and $20$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 8 λ + 0) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $λ^{2}$ and $0$ .

$p (λ) = (- λ - 2) ((- λ + 1) (λ^{2} - 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Expand $(- λ + 1) (λ^{2} - 8 λ)$ using the FOIL Method.

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Apply the distributive property.

$p (λ) = (- λ - 2) (- λ (λ^{2} - 8 λ) + 1 (λ^{2} - 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ λ^{2} - λ (- 8 λ) + 1 (λ^{2} - 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ λ^{2} - λ (- 8 λ) + (1 λ^{2} + 1 (- 8 λ)) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Remove parentheses.

$p (λ) = (- λ - 2) (- λ λ^{2} - λ (- 8 λ) + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify and combine like terms.

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Simplify each term.

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Multiply $λ$ by $λ^{2}$ by adding the exponents.

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Move $λ^{2}$ .

$p (λ) = (- λ - 2) (- (λ^{2} λ) - λ (- 8 λ) + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ^{2}$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (- λ - 2) (- λ^{2 + 1} - λ (- 8 λ) + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $2$ and $1$ .

$p (λ) = (- λ - 2) (- λ^{3} - λ (- 8 λ) + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = (- λ - 2) (- λ^{3} - λ λ \cdot - 8 + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (- λ - 2) (- λ^{3} - λ^{1 + 1} \cdot - 8 + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $1$ and $1$ .

$p (λ) = (- λ - 2) (- λ^{3} - λ^{2} \cdot - 8 + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 8$ by $- 1$ .

$p (λ) = (- λ - 2) (- λ^{3} + 8 λ^{2} + 1 λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ^{2}$ by $1$ .

$p (λ) = (- λ - 2) (- λ^{3} + 8 λ^{2} + λ^{2} + 1 (- 8 λ) - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 8 λ$ by $1$ .

$p (λ) = (- λ - 2) (- λ^{3} + 8 λ^{2} + λ^{2} - 8 λ - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $8 λ^{2}$ and $λ^{2}$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 5 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} |$ is $20 λ + 25$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 ((4) (- λ + 10) - 5 \cdot 9) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Apply the distributive property.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (4 (- λ) + 4 \cdot 10 - 5 \cdot 9) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $4$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (- 4 λ + 4 \cdot 10 - 5 \cdot 9) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $4$ by $10$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (- 4 λ + 40 - 5 \cdot 9) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $9$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (- 4 λ + 40 - 45) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by multiplying through.

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Subtract $45$ from $40$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (- 4 λ - 5) - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ - 5 (- 4 λ) - 5 \cdot - 5 - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply.

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Multiply $- 4$ by $- 5$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ - 5 \cdot - 5 - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $- 5$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 4 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |$ is $- 36 λ - 8$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 ((4) (- 4) - (- λ - 2) \cdot 9)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $- 4$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 - (- λ - 2) \cdot 9)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (λ + 2) \cdot 9)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (1 λ + 2) \cdot 9)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $- 2$ .

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (λ \cdot 9 + 2 \cdot 9))) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Move $9$ to the left of the expression $λ \cdot 9$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (9 \cdot λ + 2 \cdot 9))) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $9$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (9 \cdot λ + 18))) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $9$ by $λ$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + (9 λ + 18))) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by multiplying through.

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Remove unnecessary parentheses.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (- 16 + 9 λ + 18)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 16$ and $18$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (9 λ + 2)) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 4 (9 λ) - 4 \cdot 2) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply.

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Multiply $9$ by $- 4$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 36 λ - 4 \cdot 2) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 4$ by $2$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 8 λ + 20 λ + 25 - 36 λ - 8) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 8 λ$ and $20 λ$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} + 12 λ + 25 - 36 λ - 8) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $36 λ$ from $12 λ$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 24 λ + 25 - 8) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $8$ from $25$ .

$p (λ) = (- λ - 2) (- λ^{3} + 9 λ^{2} - 24 λ + 17) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Expand $(- λ - 2) (- λ^{3} + 9 λ^{2} - 24 λ + 17)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = - λ (- λ^{3}) - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 + (- 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17) - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify terms.

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Remove unnecessary parentheses.

$p (λ) = - λ (- λ^{3}) - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify each term.

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Multiply $λ$ by $λ^{3}$ by adding the exponents.

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Move $λ^{3}$ .

$p (λ) = - λ^{3} λ \cdot - 1 - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ^{3}$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - λ^{3 + 1} \cdot - 1 - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $3$ and $1$ .

$p (λ) = - λ^{4} \cdot - 1 - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- λ^{4} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = 1 λ^{4} - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ^{4}$ by $1$ .

$p (λ) = λ^{4} - λ (9 λ^{2}) - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $λ^{2}$ by adding the exponents.

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Move $λ^{2}$ .

$p (λ) = λ^{4} - λ^{2} λ \cdot 9 - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ^{2}$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = λ^{4} - λ^{2 + 1} \cdot 9 - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $2$ and $1$ .

$p (λ) = λ^{4} - λ^{3} \cdot 9 - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $9$ by $- 1$ .

$p (λ) = λ^{4} - 9 λ^{3} - λ (- 24 λ) - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = λ^{4} - 9 λ^{3} - λ λ \cdot - 24 - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = λ^{4} - 9 λ^{3} - λ^{1 + 1} \cdot - 24 - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $1$ and $1$ .

$p (λ) = λ^{4} - 9 λ^{3} - λ^{2} \cdot - 24 - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 24$ by $- 1$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - λ \cdot 17 - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $17$ by $- 1$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - 17 λ - 2 (- λ^{3}) - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $- 2$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - 17 λ + 2 λ^{3} - 2 (9 λ^{2}) - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $9$ by $- 2$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - 17 λ + 2 λ^{3} - 18 λ^{2} - 2 (- 24 λ) - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 24$ by $- 2$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - 17 λ + 2 λ^{3} - 18 λ^{2} + 48 λ - 2 \cdot 17 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $17$ .

$p (λ) = λ^{4} - 9 λ^{3} + 24 λ^{2} - 17 λ + 2 λ^{3} - 18 λ^{2} + 48 λ - 34 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by adding terms.

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Add $- 9 λ^{3}$ and $2 λ^{3}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 24 λ^{2} - 17 λ - 18 λ^{2} + 48 λ - 34 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $18 λ^{2}$ from $24 λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} - 17 λ + 48 λ - 34 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 17 λ$ and $48 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} | + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 12 | \begin{array}{c} 1 & - 2 & - 4 \\ 5 & - λ - 2 & - 4 \\ - 4 & 5 & - λ + 10 \end{array} |$ is $- 12 λ^{2} + 24 λ$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (1 | \begin{array}{c} - λ - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $1 | \begin{array}{c} - λ - 2 & - 4 \\ 5 & - λ + 10 \end{array} |$ is $λ^{2} - 8 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 ((- λ - 2) (- λ + 10) - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Expand $(- λ - 2) (- λ + 10)$ using the FOIL Method.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ (- λ + 10) - 2 (- λ + 10) - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ (- λ) - λ \cdot 10 - 2 (- λ + 10) - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ (- λ) - λ \cdot 10 + (- 2 (- λ) - 2 \cdot 10) - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Remove parentheses.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ (- λ) - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify and combine like terms.

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Simplify each term.

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Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ λ \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ^{1 + 1} \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $1$ and $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (- λ^{2} \cdot - 1 - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- λ^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (1 λ^{2} - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ^{2}$ by $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - λ \cdot 10 - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $10$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 10 λ - 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $- 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 10 λ + 2 λ - 2 \cdot 10 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $10$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 10 λ + 2 λ - 20 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 10 λ$ and $2 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 20 - 5 \cdot - 4 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 20 + 20 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by adding numbers.

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Add $- 20$ and $20$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ + 0 - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $λ^{2}$ and $0$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 5 | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} |$ is $- 10 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 ((- 2) (- λ + 10) - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (- 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $- 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (2 λ - 2 \cdot 10 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $10$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (2 λ - 20 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (2 λ - 20 + 20) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the expression.

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Add $- 20$ and $20$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (2 λ + 0) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $2 λ$ and $0$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 5 (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $- 5$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 4 | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |$ is $16 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 ((- 2) (- 4) - (- λ - 2) \cdot - 4)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $- 2$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 - (- λ - 2) \cdot - 4)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (λ + 2) \cdot - 4)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (1 λ + 2) \cdot - 4)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $- 2$ .

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (λ \cdot - 4 + 2 \cdot - 4))) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Move $- 4$ to the left of the expression $λ \cdot - 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (- 4 \cdot λ + 2 \cdot - 4))) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (- 4 \cdot λ - 8))) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 4$ by $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 + (- 4 λ - 8))) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the expression.

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Remove unnecessary parentheses.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (8 - 4 λ - 8)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $8$ from $8$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (- 4 λ + 0)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 4 λ$ and $0$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ - 4 (- 4 λ)) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 4$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 8 λ - 10 λ + 16 λ) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $10 λ$ from $- 8 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 18 λ + 16 λ) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 18 λ$ and $16 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 (λ^{2} - 2 λ) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} - 12 (- 2 λ) + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $- 12$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ + 6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} | - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $6 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ - 4 & 5 & - λ + 10 \end{array} |$ is $- 36 λ + 12 λ^{2} + 18$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (1 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} | - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $1 | \begin{array}{c} 4 & 9 \\ 5 & - λ + 10 \end{array} |$ is $- 4 λ - 5$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 ((4) (- λ + 10) - 5 \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (4 (- λ) + 4 \cdot 10 - 5 \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ + 4 \cdot 10 - 5 \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $4$ by $10$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ + 40 - 5 \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $9$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ + 40 - 45 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $45$ from $40$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} | - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ 5 & - λ + 10 \end{array} |$ is $2 λ^{2} - 2 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 - (- λ + 1) ((- 2) (- λ + 10) - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1 \cdot 1) ((- 2) (- λ + 10) - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (1 λ - 1 \cdot 1) ((- 2) (- λ + 10) - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) ((- 2) (- λ + 10) - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify each term.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (- 2 (- λ) - 2 \cdot 10 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 1$ by $- 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (2 λ - 2 \cdot 10 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 2$ by $10$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (2 λ - 20 - 5 \cdot - 4) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 5$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (2 λ - 20 + 20) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify by multiplying through.

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Add $- 20$ and $20$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (2 λ + 0) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $2 λ$ and $0$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + (λ - 1) (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + λ (2 λ) - 1 (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + λ λ \cdot 2 - 1 (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + λ^{1 + 1} \cdot 2 - 1 (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $1$ and $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + λ^{2} \cdot 2 - 1 (2 λ) - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + λ^{2} \cdot 2 - 2 λ - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify each term.

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Move $2$ to the left of the expression $λ^{2} \cdot 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 \cdot λ^{2} - 2 λ - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ - 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 4 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |$ is $8$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ - 4 ((- 2) (9) - 4 \cdot - 4)) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $- 2$ by $9$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ - 4 (- 18 - 4 \cdot - 4)) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 4$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ - 4 (- 18 + 16)) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the expression.

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Add $- 18$ and $16$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ - 4 \cdot - 2) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $- 4$ by $- 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 4 λ - 5 + 2 λ^{2} - 2 λ + 8) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Subtract $2 λ$ from $- 4 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 6 λ - 5 + 2 λ^{2} + 8) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Add $- 5$ and $8$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ \cdot 6 (- 6 λ + 2 λ^{2} + 3) - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify the determinant.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ + 6 (- 6 λ) + 6 (2 λ^{2}) + 6 \cdot 3 - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Simplify.

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Multiply $- 6$ by $6$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 6 (2 λ^{2}) + 6 \cdot 3 - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $2$ by $6$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 6 \cdot 3 - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

Multiply $6$ by $3$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$

The determinant of $- 3 | \begin{array}{c} 1 & - 2 & - 4 \\ - λ + 1 & 4 & 9 \\ 5 & - λ - 2 & - 4 \end{array} |$ is $- 39 λ + 12 λ^{2} + 24$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (1 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} | - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

The determinant of $1 | \begin{array}{c} 4 & 9 \\ - λ - 2 & - 4 \end{array} |$ is $9 λ + 2$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 ((4) (- 4) - (- λ - 2) \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 - (- λ - 2) \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (λ + 2) \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (1 λ + 2) \cdot 9 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $- 2$ .

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (λ \cdot 9 + 2 \cdot 9) - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Move $9$ to the left of the expression $λ \cdot 9$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (9 \cdot λ + 2 \cdot 9) - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $2$ by $9$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (9 \cdot λ + 18) - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $9$ by $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + (9 λ + 18) - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify the expression.

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Remove unnecessary parentheses.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (- 16 + 9 λ + 18 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Add $- 16$ and $18$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} | + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

The determinant of $- (- λ + 1) | \begin{array}{c} - 2 & - 4 \\ - λ - 2 & - 4 \end{array} |$ is $- 4 λ^{2} + 4 λ$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - (- λ + 1) ((- 2) (- 4) - (- λ - 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify the determinant.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1 \cdot 1) ((- 2) (- 4) - (- λ - 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (1 λ - 1 \cdot 1) ((- 2) (- 4) - (- λ - 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) ((- 2) (- 4) - (- λ - 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify each term.

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Multiply $- 2$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 - (- λ - 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (λ + 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (1 λ + 2) \cdot - 4) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $λ$ by $1$ .

Multiply $- 1$ by $- 2$ .

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (λ \cdot - 4 + 2 \cdot - 4)) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Move $- 4$ to the left of the expression $λ \cdot - 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (- 4 \cdot λ + 2 \cdot - 4)) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $2$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (- 4 \cdot λ - 8)) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $- 4$ by $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 + (- 4 λ - 8)) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify by multiplying through.

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Remove unnecessary parentheses.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (8 - 4 λ - 8) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify by subtracting numbers.

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Subtract $8$ from $8$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (- 4 λ + 0) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Add $- 4 λ$ and $0$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + (λ - 1) (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ (- 4 λ) - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $λ$ by $λ$ by adding the exponents.

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Move $λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ λ \cdot - 4 - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ λ \cdot - 4 - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Raise $λ$ to the power of $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ λ \cdot - 4 - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ^{1 + 1} \cdot - 4 - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Add $1$ and $1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ^{2} \cdot - 4 - 1 (- 4 λ) + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $- 4$ by $- 1$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 + λ^{2} \cdot - 4 + 4 λ + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Simplify each term.

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Move $- 4$ to the left of the expression $λ^{2} \cdot - 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 \cdot λ^{2} + 4 λ + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

Multiply $- 4$ by $λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ + 5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |)$

The determinant of $5 | \begin{array}{c} - 2 & - 4 \\ 4 & 9 \end{array} |$ is $- 10$ .

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The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ + 5 ((- 2) (9) - 4 \cdot - 4))$

Simplify the determinant.

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Simplify each term.

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Multiply $- 2$ by $9$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ + 5 (- 18 - 4 \cdot - 4))$

Multiply $- 4$ by $- 4$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ + 5 (- 18 + 16))$

Simplify the expression.

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Add $- 18$ and $16$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ + 5 \cdot - 2)$

Multiply $5$ by $- 2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (9 λ + 2 - 4 λ^{2} + 4 λ - 10)$

Add $9 λ$ and $4 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (13 λ + 2 - 4 λ^{2} - 10)$

Subtract $10$ from $2$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (13 λ - 4 λ^{2} - 8)$

Simplify the determinant.

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Apply the distributive property.

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 3 (13 λ) - 3 (- 4 λ^{2}) - 3 \cdot - 8$

Simplify.

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Multiply $13$ by $- 3$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 39 λ - 3 (- 4 λ^{2}) - 3 \cdot - 8$

Multiply $- 4$ by $- 3$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 39 λ + 12 λ^{2} - 3 \cdot - 8$

Multiply $- 3$ by $- 8$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 - 12 λ^{2} + 24 λ - 36 λ + 12 λ^{2} + 18 - 39 λ + 12 λ^{2} + 24$

Subtract $12 λ^{2}$ from $6 λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} - 6 λ^{2} + 31 λ - 34 + 24 λ - 36 λ + 12 λ^{2} + 18 - 39 λ + 12 λ^{2} + 24$

Add $- 6 λ^{2}$ and $12 λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 6 λ^{2} + 31 λ - 34 + 24 λ - 36 λ + 18 - 39 λ + 12 λ^{2} + 24$

Add $6 λ^{2}$ and $12 λ^{2}$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} + 31 λ - 34 + 24 λ - 36 λ + 18 - 39 λ + 24$

Add $31 λ$ and $24 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} + 55 λ - 34 - 36 λ + 18 - 39 λ + 24$

Subtract $36 λ$ from $55 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} + 19 λ - 34 + 18 - 39 λ + 24$

Subtract $39 λ$ from $19 λ$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} - 20 λ - 34 + 18 + 24$

Simplify by adding numbers.

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Add $- 34$ and $18$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} - 20 λ - 16 + 24$

Add $- 16$ and $24$ .

$p (λ) = λ^{4} - 7 λ^{3} + 18 λ^{2} - 20 λ + 8$

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