Algebra Examples

$3 {(x + 2 y)}^{2} - 13 x (x + 2 y) + 4 x^{2}$

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 13$ .

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Step 1.1.1

Reorder terms.

$4 x^{2} + 3 {(x + 2 y)}^{2} - 13 x (x + 2 y)$

Step 1.1.2

Reorder $3 {(x + 2 y)}^{2}$ and $- 13 x (x + 2 y)$ .

$4 x^{2} - 13 x (x + 2 y) + 3 {(x + 2 y)}^{2}$

Step 1.1.3

Factor $- 13$ out of $- 13 x (x + 2 y)$ .

$4 x^{2} - 13 (x (x + 2 y)) + 3 {(x + 2 y)}^{2}$

Step 1.1.4

Rewrite $- 13$ as $- 1$ plus $- 12$

$4 x^{2} + (- 1 - 12) (x (x + 2 y)) + 3 {(x + 2 y)}^{2}$

Step 1.1.5

Apply the distributive property.

$4 x^{2} - 1 (x (x + 2 y)) - 12 (x (x + 2 y)) + 3 {(x + 2 y)}^{2}$

Step 1.1.6

Remove unnecessary parentheses.

$4 x^{2} - 1 x (x + 2 y) - 12 (x (x + 2 y)) + 3 {(x + 2 y)}^{2}$

Step 1.1.7

Remove unnecessary parentheses.

$4 x^{2} - 1 x (x + 2 y) - 12 x (x + 2 y) + 3 {(x + 2 y)}^{2}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$(4 x^{2} - 1 x (x + 2 y)) - 12 x (x + 2 y) + 3 {(x + 2 y)}^{2}$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x - 1 (x + 2 y)) - 3 (x + 2 y) (4 x - (x + 2 y))$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1 (x + 2 y)$ .

$(4 x - 1 (x + 2 y)) (x - 3 (x + 2 y))$

Step 2

Apply the distributive property.

$(4 x - 1 x - 1 (2 y)) (x - 3 (x + 2 y))$

Step 3

Rewrite $- 1 x$ as $- x$ .

$(4 x - x - 1 (2 y)) (x - 3 (x + 2 y))$

Step 4

Multiply $2$ by $- 1$ .

$(4 x - x - 2 y) (x - 3 (x + 2 y))$

Step 5

Subtract $x$ from $4 x$ .

$(3 x - 2 y) (x - 3 (x + 2 y))$

Step 6

Apply the distributive property.

$(3 x - 2 y) (x - 3 x - 3 (2 y))$

Step 7

Multiply $2$ by $- 3$ .

$(3 x - 2 y) (x - 3 x - 6 y)$

Step 8

Subtract $3 x$ from $x$ .

$(3 x - 2 y) (- 2 x - 6 y)$

Step 9

Factor.

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Step 9.1

Factor $2$ out of $- 2 x - 6 y$ .

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Step 9.1.1

Factor $2$ out of $- 2 x$ .

$(3 x - 2 y) (2 (- x) - 6 y)$

Step 9.1.2

Factor $2$ out of $- 6 y$ .

$(3 x - 2 y) (2 (- x) + 2 (- 3 y))$

Step 9.1.3

Factor $2$ out of $2 (- x) + 2 (- 3 y)$ .

$(3 x - 2 y) (2 (- x - 3 y))$

Step 9.2

Remove unnecessary parentheses.

$(3 x - 2 y) \cdot 2 (- x - 3 y)$

Step 10

Factor.

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Step 10.1

Factor $- 1$ out of $- x - 3 y$ .

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Step 10.1.1

Factor $- 1$ out of $- x$ .

$(3 x - 2 y) \cdot 2 (- (x) - 3 y)$

Step 10.1.2

Factor $- 1$ out of $- 3 y$ .

$(3 x - 2 y) \cdot 2 (- (x) - (3 y))$

Step 10.1.3

Factor $- 1$ out of $- (x) - (3 y)$ .

$(3 x - 2 y) \cdot 2 (- (x + 3 y))$

Step 10.2

Remove unnecessary parentheses.

$(3 x - 2 y) \cdot 2 \cdot - 1 (x + 3 y)$

Step 11

Combine exponents.

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Step 11.1

Factor out negative.

$- ((3 x - 2 y) \cdot 2 (x + 3 y))$

Step 11.2

Multiply $2$ by $- 1$ .

$- 2 ((3 x - 2 y) (x + 3 y))$

Step 12

Remove unnecessary parentheses.

$- 2 (3 x - 2 y) (x + 3 y)$