Calculus Examples

$f (x) = - 36 {(x + 4)}^{2} (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)$

Step 1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$\frac{d}{d x} [- 36 ((x + 4) (x + 4)) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [- 36 (x (x + 4) + 4 (x + 4)) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [- 36 (x \cdot x + x \cdot 4 + 4 (x + 4)) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [- 36 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [- 36 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 3.1.2

Move $4$ to the left of $x$ .

$\frac{d}{d x} [- 36 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d x} [- 36 (x^{2} + 4 x + 4 x + 16) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 3.2

Add $4 x$ and $4 x$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 {(2 x - 3)}^{2} (x + 4)]$

Step 4

Rewrite ${(2 x - 3)}^{2}$ as $(2 x - 3) (2 x - 3)$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 ((2 x - 3) (2 x - 3)) (x + 4)]$

Step 5

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 x (2 x - 3) - 3 (2 x - 3)) (x + 4)]$

Step 5.2

Apply the distributive property.

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3)) (x + 4)]$

Step 5.3

Apply the distributive property.

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.2.1

Move $x$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6.1.4

Multiply $- 3$ by $2$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3) (x + 4)]$

Step 6.1.5

Multiply $2$ by $- 3$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} - 6 x - 6 x - 3 \cdot - 3) (x + 4)]$

Step 6.1.6

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} - 6 x - 6 x + 9) (x + 4)]$

Step 6.2

Subtract $6 x$ from $- 6 x$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 7

By the Sum Rule, the derivative of $- 36 (x^{2} + 8 x + 16) (2 x - 3) + 18 (4 x^{2} - 12 x + 9) (x + 4)$ with respect to $x$ is $\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3)] + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$ .

$\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3)] + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8

Evaluate $\frac{d}{d x} [- 36 (x^{2} + 8 x + 16) (2 x - 3)]$ .

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Step 8.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 (x^{2} + 8 x + 16) (2 x - 3)$ with respect to $x$ is $- 36 \frac{d}{d x} [(x^{2} + 8 x + 16) (2 x - 3)]$ .

$- 36 \frac{d}{d x} [(x^{2} + 8 x + 16) (2 x - 3)] + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 8 x + 16$ and $g (x) = 2 x - 3$ .

$- 36 ((x^{2} + 8 x + 16) \frac{d}{d x} [2 x - 3] + (2 x - 3) \frac{d}{d x} [x^{2} + 8 x + 16]) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.3

By the Sum Rule, the derivative of $2 x - 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

$- 36 ((x^{2} + 8 x + 16) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]) + (2 x - 3) \frac{d}{d x} [x^{2} + 8 x + 16]) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.4

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 36 ((x^{2} + 8 x + 16) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (2 x - 3) \frac{d}{d x} [x^{2} + 8 x + 16]) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + \frac{d}{d x} [- 3]) + (2 x - 3) \frac{d}{d x} [x^{2} + 8 x + 16]) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) \frac{d}{d x} [x^{2} + 8 x + 16]) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.7

By the Sum Rule, the derivative of $x^{2} + 8 x + 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [16]$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [16])) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) (2 x + \frac{d}{d x} [8 x] + \frac{d}{d x} [16])) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.9

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) (2 x + 8 \frac{d}{d x} [x] + \frac{d}{d x} [16])) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) (2 x + 8 \cdot 1 + \frac{d}{d x} [16])) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.11

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$- 36 ((x^{2} + 8 x + 16) (2 \cdot 1 + 0) + (2 x - 3) (2 x + 8 \cdot 1 + 0)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.12

Multiply $2$ by $1$ .

$- 36 ((x^{2} + 8 x + 16) (2 + 0) + (2 x - 3) (2 x + 8 \cdot 1 + 0)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.13

Add $2$ and $0$ .

$- 36 ((x^{2} + 8 x + 16) \cdot 2 + (2 x - 3) (2 x + 8 \cdot 1 + 0)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.14

Move $2$ to the left of $x^{2} + 8 x + 16$ .

$- 36 (2 \cdot (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8 \cdot 1 + 0)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.15

Multiply $8$ by $1$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8 + 0)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 8.16

Add $2 x + 8$ and $0$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + \frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$

Step 9

Evaluate $\frac{d}{d x} [18 (4 x^{2} - 12 x + 9) (x + 4)]$ .

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Step 9.1

Since $18$ is constant with respect to $x$ , the derivative of $18 (4 x^{2} - 12 x + 9) (x + 4)$ with respect to $x$ is $18 \frac{d}{d x} [(4 x^{2} - 12 x + 9) (x + 4)]$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 \frac{d}{d x} [(4 x^{2} - 12 x + 9) (x + 4)]$

Step 9.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 4 x^{2} - 12 x + 9$ and $g (x) = x + 4$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) \frac{d}{d x} [x + 4] + (x + 4) \frac{d}{d x} [4 x^{2} - 12 x + 9])$

Step 9.3

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (\frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [4 x^{2} - 12 x + 9])$

Step 9.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [4 x^{2} - 12 x + 9])$

Step 9.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) \frac{d}{d x} [4 x^{2} - 12 x + 9])$

Step 9.6

By the Sum Rule, the derivative of $4 x^{2} - 12 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]))$

Step 9.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]))$

Step 9.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (4 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [9]))$

Step 9.9

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (4 (2 x) - 12 \frac{d}{d x} [x] + \frac{d}{d x} [9]))$

Step 9.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (4 (2 x) - 12 \cdot 1 + \frac{d}{d x} [9]))$

Step 9.11

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) (1 + 0) + (x + 4) (4 (2 x) - 12 \cdot 1 + 0))$

Step 9.12

Add $1$ and $0$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 ((4 x^{2} - 12 x + 9) \cdot 1 + (x + 4) (4 (2 x) - 12 \cdot 1 + 0))$

Step 9.13

Multiply $4 x^{2} - 12 x + 9$ by $1$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (4 (2 x) - 12 \cdot 1 + 0))$

Step 9.14

Multiply $2$ by $4$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (8 x - 12 \cdot 1 + 0))$

Step 9.15

Multiply $- 12$ by $1$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (8 x - 12 + 0))$

Step 9.16

Add $8 x - 12$ and $0$ .

$- 36 (2 (x^{2} + 8 x + 16) + (2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (8 x - 12))$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- 36 (2 x^{2} + 2 (8 x) + 2 \cdot 16 + (2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (8 x - 12))$

Step 10.2

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 ((2 x - 3) (2 x + 8)) + 18 (4 x^{2} - 12 x + 9 + (x + 4) (8 x - 12))$

Step 10.3

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 ((2 x - 3) (2 x + 8)) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x - 12))$

Step 10.4

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 ((2 x - 3) (2 x) + (2 x - 3) \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x - 12))$

Step 10.5

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x) - 3 (2 x) + (2 x - 3) \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x - 12))$

Step 10.6

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x) - 3 (2 x) + 2 x \cdot 8 - 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x - 12))$

Step 10.7

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x - 12))$

Step 10.8

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 ((x + 4) (8 x) + (x + 4) \cdot - 12)$

Step 10.9

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x) + 4 (8 x) + (x + 4) \cdot - 12)$

Step 10.10

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x) + 4 (8 x) + x \cdot - 12 + 4 \cdot - 12)$

Step 10.11

Apply the distributive property.

$- 36 (2 x^{2}) - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12

Combine terms.

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Step 10.12.1

Multiply $2$ by $- 36$ .

$- 72 x^{2} - 36 (2 (8 x)) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.2

Multiply $8$ by $2$ .

$- 72 x^{2} - 36 (16 x) - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.3

Multiply $16$ by $- 36$ .

$- 72 x^{2} - 576 x - 36 (2 \cdot 16) - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.4

Multiply $2$ by $16$ .

$- 72 x^{2} - 576 x - 36 \cdot 32 - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.5

Multiply $- 36$ by $32$ .

$- 72 x^{2} - 576 x - 1152 - 36 (2 x (2 x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.6

Multiply $2$ by $2$ .

$- 72 x^{2} - 576 x - 1152 - 36 (4 x \cdot x) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.7

Raise $x$ to the power of $1$ .

$- 72 x^{2} - 576 x - 1152 - 36 (4 (x^{1} x)) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.8

Raise $x$ to the power of $1$ .

$- 72 x^{2} - 576 x - 1152 - 36 (4 (x^{1} x^{1})) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.9

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

$- 72 x^{2} - 576 x - 1152 - 36 (4 x^{1 + 1}) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.10

Add $1$ and $1$ .

$- 72 x^{2} - 576 x - 1152 - 36 (4 x^{2}) - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.11

Multiply $4$ by $- 36$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} - 36 (- 3 (2 x)) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.12

Multiply $2$ by $- 3$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} - 36 (- 6 x) - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.13

Multiply $- 6$ by $- 36$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} + 216 x - 36 (2 x \cdot 8) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.14

Multiply $8$ by $2$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} + 216 x - 36 (16 x) - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.15

Multiply $16$ by $- 36$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} + 216 x - 576 x - 36 (- 3 \cdot 8) + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.16

Multiply $- 3$ by $8$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} + 216 x - 576 x - 36 \cdot - 24 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.17

Multiply $- 36$ by $- 24$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} + 216 x - 576 x + 864 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.18

Subtract $576 x$ from $216 x$ .

$- 72 x^{2} - 576 x - 1152 - 144 x^{2} - 360 x + 864 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.19

Subtract $144 x^{2}$ from $- 72 x^{2}$ .

$- 216 x^{2} - 576 x - 1152 - 360 x + 864 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.20

Subtract $360 x$ from $- 576 x$ .

$- 216 x^{2} - 936 x - 1152 + 864 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.21

Add $- 1152$ and $864$ .

$- 216 x^{2} - 936 x - 288 + 18 (4 x^{2}) + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.22

Multiply $4$ by $18$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} + 18 (- 12 x) + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.23

Multiply $- 12$ by $18$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 18 \cdot 9 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.24

Multiply $18$ by $9$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 18 (x (8 x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.25

Raise $x$ to the power of $1$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 18 (8 (x^{1} x)) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.26

Raise $x$ to the power of $1$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 18 (8 (x^{1} x^{1})) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.27

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

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 18 (8 x^{1 + 1}) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.28

Add $1$ and $1$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 18 (8 x^{2}) + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.29

Multiply $8$ by $18$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 18 (4 (8 x)) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.30

Multiply $8$ by $4$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 18 (32 x) + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.31

Multiply $32$ by $18$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 576 x + 18 (x \cdot - 12) + 18 (4 \cdot - 12)$

Step 10.12.32

Move $- 12$ to the left of $x$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 576 x + 18 (- 12 \cdot x) + 18 (4 \cdot - 12)$

Step 10.12.33

Multiply $- 12$ by $18$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 576 x - 216 x + 18 (4 \cdot - 12)$

Step 10.12.34

Multiply $4$ by $- 12$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 576 x - 216 x + 18 \cdot - 48$

Step 10.12.35

Multiply $18$ by $- 48$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 576 x - 216 x - 864$

Step 10.12.36

Subtract $216 x$ from $576 x$ .

$- 216 x^{2} - 936 x - 288 + 72 x^{2} - 216 x + 162 + 144 x^{2} + 360 x - 864$

Step 10.12.37

Add $72 x^{2}$ and $144 x^{2}$ .

$- 216 x^{2} - 936 x - 288 + 216 x^{2} - 216 x + 162 + 360 x - 864$

Step 10.12.38

Add $- 216 x$ and $360 x$ .

$- 216 x^{2} - 936 x - 288 + 216 x^{2} + 144 x + 162 - 864$

Step 10.12.39

Subtract $864$ from $162$ .

$- 216 x^{2} - 936 x - 288 + 216 x^{2} + 144 x - 702$

Step 10.12.40

Add $- 216 x^{2}$ and $216 x^{2}$ .

$- 936 x - 288 + 0 + 144 x - 702$

Step 10.12.41

Add $- 936 x - 288$ and $0$ .

$- 936 x - 288 + 144 x - 702$

Step 10.12.42

Add $- 936 x$ and $144 x$ .

$- 792 x - 288 - 702$

Step 10.12.43

Subtract $702$ from $- 288$ .

$- 792 x - 990$