Calculus Examples

$f (x) = 4 {(x - 5)}^{4} (3 x + 1) (x^{2} - 4)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 {(x - 5)}^{4} (3 x + 1) (x^{2} - 4)$ with respect to $x$ is $4 \frac{d}{d x} [({(x - 5)}^{4} (3 x + 1)) (x^{2} - 4)]$ .

$4 \frac{d}{d x} [({(x - 5)}^{4} (3 x + 1)) (x^{2} - 4)]$

Step 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 $f (x) = {(x - 5)}^{4} (3 x + 1)$ and $g (x) = x^{2} - 4$ .

$4 ({(x - 5)}^{4} (3 x + 1) \frac{d}{d x} [x^{2} - 4] + (x^{2} - 4) \frac{d}{d x} [{(x - 5)}^{4} (3 x + 1)])$

Step 3

Differentiate.

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Step 3.1

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

$4 ({(x - 5)}^{4} (3 x + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]) + (x^{2} - 4) \frac{d}{d x} [{(x - 5)}^{4} (3 x + 1)])$

Step 3.2

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

$4 ({(x - 5)}^{4} (3 x + 1) (2 x + \frac{d}{d x} [- 4]) + (x^{2} - 4) \frac{d}{d x} [{(x - 5)}^{4} (3 x + 1)])$

Step 3.3

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

$4 ({(x - 5)}^{4} (3 x + 1) (2 x + 0) + (x^{2} - 4) \frac{d}{d x} [{(x - 5)}^{4} (3 x + 1)])$

Step 3.4

Add $2 x$ and $0$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) \frac{d}{d x} [{(x - 5)}^{4} (3 x + 1)])$

Step 4

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 $f (x) = {(x - 5)}^{4}$ and $g (x) = 3 x + 1$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} \frac{d}{d x} [3 x + 1] + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} (\frac{d}{d x} [3 x] + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} (3 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.3

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

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} (3 \cdot 1 + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.4

Multiply $3$ by $1$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} (3 + \frac{d}{d x} [1]) + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} (3 + 0) + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $3$ and $0$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) ({(x - 5)}^{4} \cdot 3 + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 5.6.2

Move $3$ to the left of ${(x - 5)}^{4}$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 \cdot {(x - 5)}^{4} + (3 x + 1) \frac{d}{d x} [{(x - 5)}^{4}]))$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = x - 5$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + (3 x + 1) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [x - 5])))$

Step 6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + (3 x + 1) (4 u^{3} \frac{d}{d x} [x - 5])))$

Step 6.3

Replace all occurrences of $u$ with $x - 5$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + (3 x + 1) (4 {(x - 5)}^{3} \frac{d}{d x} [x - 5])))$

Step 7

Differentiate.

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Step 7.1

Move $4$ to the left of $3 x + 1$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 \cdot (3 x + 1) {(x - 5)}^{3} \frac{d}{d x} [x - 5]))$

Step 7.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 (3 x + 1) {(x - 5)}^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])))$

Step 7.3

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

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 (3 x + 1) {(x - 5)}^{3} (1 + \frac{d}{d x} [- 5])))$

Step 7.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 (3 x + 1) {(x - 5)}^{3} (1 + 0)))$

Step 7.5

Simplify the expression.

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Step 7.5.1

Add $1$ and $0$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 (3 x + 1) {(x - 5)}^{3} \cdot 1))$

Step 7.5.2

Multiply $4$ by $1$ .

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + 4 (3 x + 1) {(x - 5)}^{3}))$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$4 ({(x - 5)}^{4} (3 x + 1) (2 x) + (x^{2} - 4) (3 {(x - 5)}^{4} + (4 (3 x) + 4 \cdot 1) {(x - 5)}^{3}))$

Step 8.2

Apply the distributive property.

$4 ({(x - 5)}^{4} (3 x + 1) (2 x)) + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (4 (3 x) + 4 \cdot 1) {(x - 5)}^{3}))$

Step 8.3

Multiply $2$ by $4$ .

$8 ({(x - 5)}^{4} (3 x + 1) (x)) + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (4 (3 x) + 4 \cdot 1) {(x - 5)}^{3}))$

Step 8.4

Multiply $3$ by $4$ .

$8 {(x - 5)}^{4} (3 x + 1) x + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4 \cdot 1) {(x - 5)}^{3}))$

Step 8.5

Multiply $4$ by $1$ .

$8 {(x - 5)}^{4} (3 x + 1) x + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.6

Factor $4$ out of $8 {(x - 5)}^{4} (3 x + 1) x + 4 (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3})$ .

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Step 8.6.1

Factor $4$ out of $8 {(x - 5)}^{4} (3 x + 1) x$ .

$4 (2 {(x - 5)}^{4} (3 x + 1) x) + 4 (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3})$

Step 8.6.2

Factor $4$ out of $4 (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3})$ .

$4 (2 {(x - 5)}^{4} (3 x + 1) x) + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.6.3

Factor $4$ out of $4 (2 {(x - 5)}^{4} (3 x + 1) x) + 4 ((x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$ .

$4 (2 {(x - 5)}^{4} (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7

Simplify each term.

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Step 8.7.1

Use the Binomial Theorem.

$4 (2 (x^{4} + 4 x^{3} \cdot - 5 + 6 x^{2} {(- 5)}^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2

Simplify each term.

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Step 8.7.2.1

Multiply $- 5$ by $4$ .

$4 (2 (x^{4} - 20 x^{3} + 6 x^{2} {(- 5)}^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2.2

Raise $- 5$ to the power of $2$ .

$4 (2 (x^{4} - 20 x^{3} + 6 x^{2} \cdot 25 + 4 x {(- 5)}^{3} + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2.3

Multiply $25$ by $6$ .

$4 (2 (x^{4} - 20 x^{3} + 150 x^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2.4

Raise $- 5$ to the power of $3$ .

$4 (2 (x^{4} - 20 x^{3} + 150 x^{2} + 4 x \cdot - 125 + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2.5

Multiply $- 125$ by $4$ .

$4 (2 (x^{4} - 20 x^{3} + 150 x^{2} - 500 x + {(- 5)}^{4}) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.2.6

Raise $- 5$ to the power of $4$ .

$4 (2 (x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.3

Apply the distributive property.

$4 ((2 x^{4} + 2 (- 20 x^{3}) + 2 (150 x^{2}) + 2 (- 500 x) + 2 \cdot 625) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.4

Simplify.

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Step 8.7.4.1

Multiply $- 20$ by $2$ .

$4 ((2 x^{4} - 40 x^{3} + 2 (150 x^{2}) + 2 (- 500 x) + 2 \cdot 625) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.4.2

Multiply $150$ by $2$ .

$4 ((2 x^{4} - 40 x^{3} + 300 x^{2} + 2 (- 500 x) + 2 \cdot 625) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.4.3

Multiply $- 500$ by $2$ .

$4 ((2 x^{4} - 40 x^{3} + 300 x^{2} - 1000 x + 2 \cdot 625) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.4.4

Multiply $2$ by $625$ .

$4 ((2 x^{4} - 40 x^{3} + 300 x^{2} - 1000 x + 1250) (3 x + 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.5

Expand $(2 x^{4} - 40 x^{3} + 300 x^{2} - 1000 x + 1250) (3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$4 ((2 x^{4} (3 x) + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6

Simplify each term.

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Step 8.7.6.1

Rewrite using the commutative property of multiplication.

$4 ((2 \cdot 3 x^{4} x + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.2

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 8.7.6.2.1

Move $x$ .

$4 ((2 \cdot 3 (x \cdot x^{4}) + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.2.2

Multiply $x$ by $x^{4}$ .

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Step 8.7.6.2.2.1

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

$4 ((2 \cdot 3 (x^{1} x^{4}) + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.2.2.2

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

$4 ((2 \cdot 3 x^{1 + 4} + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.2.3

Add $1$ and $4$ .

$4 ((2 \cdot 3 x^{5} + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.3

Multiply $2$ by $3$ .

$4 ((6 x^{5} + 2 x^{4} \cdot 1 - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.4

Multiply $2$ by $1$ .

$4 ((6 x^{5} + 2 x^{4} - 40 x^{3} (3 x) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.5

Rewrite using the commutative property of multiplication.

$4 ((6 x^{5} + 2 x^{4} - 40 \cdot 3 x^{3} x - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.6

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

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Step 8.7.6.6.1

Move $x$ .

$4 ((6 x^{5} + 2 x^{4} - 40 \cdot 3 (x \cdot x^{3}) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.6.2

Multiply $x$ by $x^{3}$ .

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Step 8.7.6.6.2.1

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

$4 ((6 x^{5} + 2 x^{4} - 40 \cdot 3 (x^{1} x^{3}) - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.6.2.2

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

$4 ((6 x^{5} + 2 x^{4} - 40 \cdot 3 x^{1 + 3} - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.6.3

Add $1$ and $3$ .

$4 ((6 x^{5} + 2 x^{4} - 40 \cdot 3 x^{4} - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.7

Multiply $- 40$ by $3$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} \cdot 1 + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.8

Multiply $- 40$ by $1$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 x^{2} (3 x) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.9

Rewrite using the commutative property of multiplication.

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 \cdot 3 x^{2} x + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.10

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

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Step 8.7.6.10.1

Move $x$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 \cdot 3 (x \cdot x^{2}) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.10.2

Multiply $x$ by $x^{2}$ .

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Step 8.7.6.10.2.1

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

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 \cdot 3 (x^{1} x^{2}) + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.10.2.2

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

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 \cdot 3 x^{1 + 2} + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.10.3

Add $1$ and $2$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 300 \cdot 3 x^{3} + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.11

Multiply $300$ by $3$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} \cdot 1 - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.12

Multiply $300$ by $1$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 1000 x (3 x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.13

Rewrite using the commutative property of multiplication.

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 1000 \cdot 3 x \cdot x - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.14

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

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Step 8.7.6.14.1

Move $x$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 1000 \cdot 3 (x \cdot x) - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.14.2

Multiply $x$ by $x$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 1000 \cdot 3 x^{2} - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.15

Multiply $- 1000$ by $3$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x \cdot 1 + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.16

Multiply $- 1000$ by $1$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x + 1250 (3 x) + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.17

Multiply $3$ by $1250$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x + 3750 x + 1250 \cdot 1) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.6.18

Multiply $1250$ by $1$ .

$4 ((6 x^{5} + 2 x^{4} - 120 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x + 3750 x + 1250) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.7

Subtract $120 x^{4}$ from $2 x^{4}$ .

$4 ((6 x^{5} - 118 x^{4} - 40 x^{3} + 900 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x + 3750 x + 1250) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.8

Add $- 40 x^{3}$ and $900 x^{3}$ .

$4 ((6 x^{5} - 118 x^{4} + 860 x^{3} + 300 x^{2} - 3000 x^{2} - 1000 x + 3750 x + 1250) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.9

Subtract $3000 x^{2}$ from $300 x^{2}$ .

$4 ((6 x^{5} - 118 x^{4} + 860 x^{3} - 2700 x^{2} - 1000 x + 3750 x + 1250) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.10

Add $- 1000 x$ and $3750 x$ .

$4 ((6 x^{5} - 118 x^{4} + 860 x^{3} - 2700 x^{2} + 2750 x + 1250) x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.11

Apply the distributive property.

$4 (6 x^{5} x - 118 x^{4} x + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12

Simplify.

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Step 8.7.12.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 8.7.12.1.1

Move $x$ .

$4 (6 (x \cdot x^{5}) - 118 x^{4} x + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.1.2

Multiply $x$ by $x^{5}$ .

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Step 8.7.12.1.2.1

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

$4 (6 (x^{1} x^{5}) - 118 x^{4} x + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.1.2.2

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

$4 (6 x^{1 + 5} - 118 x^{4} x + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.1.3

Add $1$ and $5$ .

$4 (6 x^{6} - 118 x^{4} x + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.2

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 8.7.12.2.1

Move $x$ .

$4 (6 x^{6} - 118 (x \cdot x^{4}) + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.2.2

Multiply $x$ by $x^{4}$ .

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Step 8.7.12.2.2.1

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

$4 (6 x^{6} - 118 (x^{1} x^{4}) + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.2.2.2

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

$4 (6 x^{6} - 118 x^{1 + 4} + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.2.3

Add $1$ and $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{3} x - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.3

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

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Step 8.7.12.3.1

Move $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 (x \cdot x^{3}) - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.3.2

Multiply $x$ by $x^{3}$ .

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Step 8.7.12.3.2.1

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

$4 (6 x^{6} - 118 x^{5} + 860 (x^{1} x^{3}) - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.3.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{1 + 3} - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.3.3

Add $1$ and $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{2} x + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.4

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

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Step 8.7.12.4.1

Move $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 (x \cdot x^{2}) + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.4.2

Multiply $x$ by $x^{2}$ .

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Step 8.7.12.4.2.1

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 (x^{1} x^{2}) + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.4.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{1 + 2} + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.4.3

Add $1$ and $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x \cdot x + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.5

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

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Step 8.7.12.5.1

Move $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 (x \cdot x) + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.12.5.2

Multiply $x$ by $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 {(x - 5)}^{4} + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13

Simplify each term.

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Step 8.7.13.1

Use the Binomial Theorem.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} + 4 x^{3} \cdot - 5 + 6 x^{2} {(- 5)}^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2

Simplify each term.

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Step 8.7.13.2.1

Multiply $- 5$ by $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 6 x^{2} {(- 5)}^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2.2

Raise $- 5$ to the power of $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 6 x^{2} \cdot 25 + 4 x {(- 5)}^{3} + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2.3

Multiply $25$ by $6$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 150 x^{2} + 4 x {(- 5)}^{3} + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2.4

Raise $- 5$ to the power of $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 150 x^{2} + 4 x \cdot - 125 + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2.5

Multiply $- 125$ by $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 150 x^{2} - 500 x + {(- 5)}^{4}) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.2.6

Raise $- 5$ to the power of $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 (x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625) + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.3

Apply the distributive property.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} + 3 (- 20 x^{3}) + 3 (150 x^{2}) + 3 (- 500 x) + 3 \cdot 625 + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.4

Simplify.

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Step 8.7.13.4.1

Multiply $- 20$ by $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 3 (150 x^{2}) + 3 (- 500 x) + 3 \cdot 625 + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.4.2

Multiply $150$ by $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} + 3 (- 500 x) + 3 \cdot 625 + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.4.3

Multiply $- 500$ by $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 3 \cdot 625 + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.4.4

Multiply $3$ by $625$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) {(x - 5)}^{3}))$

Step 8.7.13.5

Use the Binomial Theorem.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) (x^{3} + 3 x^{2} \cdot - 5 + 3 x {(- 5)}^{2} + {(- 5)}^{3})))$

Step 8.7.13.6

Simplify each term.

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Step 8.7.13.6.1

Multiply $- 5$ by $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) (x^{3} - 15 x^{2} + 3 x {(- 5)}^{2} + {(- 5)}^{3})))$

Step 8.7.13.6.2

Raise $- 5$ to the power of $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) (x^{3} - 15 x^{2} + 3 x \cdot 25 + {(- 5)}^{3})))$

Step 8.7.13.6.3

Multiply $25$ by $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) (x^{3} - 15 x^{2} + 75 x + {(- 5)}^{3})))$

Step 8.7.13.6.4

Raise $- 5$ to the power of $3$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + (12 x + 4) (x^{3} - 15 x^{2} + 75 x - 125)))$

Step 8.7.13.7

Expand $(12 x + 4) (x^{3} - 15 x^{2} + 75 x - 125)$ by multiplying each term in the first expression by each term in the second expression.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x \cdot x^{3} + 12 x (- 15 x^{2}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8

Simplify each term.

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Step 8.7.13.8.1

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

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Step 8.7.13.8.1.1

Move $x^{3}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 (x^{3} x) + 12 x (- 15 x^{2}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.1.2

Multiply $x^{3}$ by $x$ .

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Step 8.7.13.8.1.2.1

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 (x^{3} x^{1}) + 12 x (- 15 x^{2}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.1.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{3 + 1} + 12 x (- 15 x^{2}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.1.3

Add $3$ and $1$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 x (- 15 x^{2}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.2

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 \cdot - 15 x \cdot x^{2} + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.3

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

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Step 8.7.13.8.3.1

Move $x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 \cdot - 15 (x^{2} x) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.3.2

Multiply $x^{2}$ by $x$ .

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Step 8.7.13.8.3.2.1

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 \cdot - 15 (x^{2} x^{1}) + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.3.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 \cdot - 15 x^{2 + 1} + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.3.3

Add $2$ and $1$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} + 12 \cdot - 15 x^{3} + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.4

Multiply $12$ by $- 15$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 12 x (75 x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.5

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 12 \cdot 75 x \cdot x + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.6

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

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Step 8.7.13.8.6.1

Move $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 12 \cdot 75 (x \cdot x) + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.6.2

Multiply $x$ by $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 12 \cdot 75 x^{2} + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.7

Multiply $12$ by $75$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 900 x^{2} + 12 x \cdot - 125 + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.8

Multiply $- 125$ by $12$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 900 x^{2} - 1500 x + 4 x^{3} + 4 (- 15 x^{2}) + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.9

Multiply $- 15$ by $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 900 x^{2} - 1500 x + 4 x^{3} - 60 x^{2} + 4 (75 x) + 4 \cdot - 125))$

Step 8.7.13.8.10

Multiply $75$ by $4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 180 x^{3} + 900 x^{2} - 1500 x + 4 x^{3} - 60 x^{2} + 300 x + 4 \cdot - 125))$

Step 8.7.13.8.11

Multiply $4$ by $- 125$ .

Step 8.7.13.9

Add $- 180 x^{3}$ and $4 x^{3}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 176 x^{3} + 900 x^{2} - 1500 x - 60 x^{2} + 300 x - 500))$

Step 8.7.13.10

Subtract $60 x^{2}$ from $900 x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 176 x^{3} + 840 x^{2} - 1500 x + 300 x - 500))$

Step 8.7.13.11

Add $- 1500 x$ and $300 x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (3 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 + 12 x^{4} - 176 x^{3} + 840 x^{2} - 1200 x - 500))$

Step 8.7.14

Add $3 x^{4}$ and $12 x^{4}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (15 x^{4} - 60 x^{3} + 450 x^{2} - 1500 x + 1875 - 176 x^{3} + 840 x^{2} - 1200 x - 500))$

Step 8.7.15

Subtract $176 x^{3}$ from $- 60 x^{3}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (15 x^{4} - 236 x^{3} + 450 x^{2} - 1500 x + 1875 + 840 x^{2} - 1200 x - 500))$

Step 8.7.16

Add $450 x^{2}$ and $840 x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (15 x^{4} - 236 x^{3} + 1290 x^{2} - 1500 x + 1875 - 1200 x - 500))$

Step 8.7.17

Subtract $1200 x$ from $- 1500 x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (15 x^{4} - 236 x^{3} + 1290 x^{2} - 2700 x + 1875 - 500))$

Step 8.7.18

Subtract $500$ from $1875$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + (x^{2} - 4) (15 x^{4} - 236 x^{3} + 1290 x^{2} - 2700 x + 1375))$

Step 8.7.19

Expand $(x^{2} - 4) (15 x^{4} - 236 x^{3} + 1290 x^{2} - 2700 x + 1375)$ by multiplying each term in the first expression by each term in the second expression.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + x^{2} (15 x^{4}) + x^{2} (- 236 x^{3}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20

Simplify each term.

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Step 8.7.20.1

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{2} x^{4} + x^{2} (- 236 x^{3}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 8.7.20.2.1

Move $x^{4}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 (x^{4} x^{2}) + x^{2} (- 236 x^{3}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{4 + 2} + x^{2} (- 236 x^{3}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.2.3

Add $4$ and $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} + x^{2} (- 236 x^{3}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.3

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{2} x^{3} + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.4

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 8.7.20.4.1

Move $x^{3}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 (x^{3} x^{2}) + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.4.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{3 + 2} + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.4.3

Add $3$ and $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + x^{2} (1290 x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.5

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{2} x^{2} + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.6

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

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Step 8.7.20.6.1

Move $x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 (x^{2} x^{2}) + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.6.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{2 + 2} + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.6.3

Add $2$ and $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} + x^{2} (- 2700 x) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.7

Rewrite using the commutative property of multiplication.

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{2} x + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.8

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

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Step 8.7.20.8.1

Move $x$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 (x \cdot x^{2}) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.8.2

Multiply $x$ by $x^{2}$ .

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Step 8.7.20.8.2.1

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 (x^{1} x^{2}) + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.8.2.2

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

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{1 + 2} + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.8.3

Add $1$ and $2$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + x^{2} \cdot 1375 - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.9

Move $1375$ to the left of $x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 \cdot x^{2} - 4 (15 x^{4}) - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.10

Multiply $15$ by $- 4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 x^{2} - 60 x^{4} - 4 (- 236 x^{3}) - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.11

Multiply $- 236$ by $- 4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 x^{2} - 60 x^{4} + 944 x^{3} - 4 (1290 x^{2}) - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.12

Multiply $1290$ by $- 4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 x^{2} - 60 x^{4} + 944 x^{3} - 5160 x^{2} - 4 (- 2700 x) - 4 \cdot 1375)$

Step 8.7.20.13

Multiply $- 2700$ by $- 4$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 x^{2} - 60 x^{4} + 944 x^{3} - 5160 x^{2} + 10800 x - 4 \cdot 1375)$

Step 8.7.20.14

Multiply $- 4$ by $1375$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1290 x^{4} - 2700 x^{3} + 1375 x^{2} - 60 x^{4} + 944 x^{3} - 5160 x^{2} + 10800 x - 5500)$

Step 8.7.21

Subtract $60 x^{4}$ from $1290 x^{4}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1230 x^{4} - 2700 x^{3} + 1375 x^{2} + 944 x^{3} - 5160 x^{2} + 10800 x - 5500)$

Step 8.7.22

Add $- 2700 x^{3}$ and $944 x^{3}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1230 x^{4} - 1756 x^{3} + 1375 x^{2} - 5160 x^{2} + 10800 x - 5500)$

Step 8.7.23

Subtract $5160 x^{2}$ from $1375 x^{2}$ .

$4 (6 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 15 x^{6} - 236 x^{5} + 1230 x^{4} - 1756 x^{3} - 3785 x^{2} + 10800 x - 5500)$

Step 8.8

Add $6 x^{6}$ and $15 x^{6}$ .

$4 (21 x^{6} - 118 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x - 236 x^{5} + 1230 x^{4} - 1756 x^{3} - 3785 x^{2} + 10800 x - 5500)$

Step 8.9

Subtract $236 x^{5}$ from $- 118 x^{5}$ .

$4 (21 x^{6} - 354 x^{5} + 860 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x + 1230 x^{4} - 1756 x^{3} - 3785 x^{2} + 10800 x - 5500)$

Step 8.10

Add $860 x^{4}$ and $1230 x^{4}$ .

$4 (21 x^{6} - 354 x^{5} + 2090 x^{4} - 2700 x^{3} + 2750 x^{2} + 1250 x - 1756 x^{3} - 3785 x^{2} + 10800 x - 5500)$

Step 8.11

Subtract $1756 x^{3}$ from $- 2700 x^{3}$ .

$4 (21 x^{6} - 354 x^{5} + 2090 x^{4} - 4456 x^{3} + 2750 x^{2} + 1250 x - 3785 x^{2} + 10800 x - 5500)$

Step 8.12

Subtract $3785 x^{2}$ from $2750 x^{2}$ .

$4 (21 x^{6} - 354 x^{5} + 2090 x^{4} - 4456 x^{3} - 1035 x^{2} + 1250 x + 10800 x - 5500)$

Step 8.13

Add $1250 x$ and $10800 x$ .

$4 (21 x^{6} - 354 x^{5} + 2090 x^{4} - 4456 x^{3} - 1035 x^{2} + 12050 x - 5500)$

Step 8.14

Apply the distributive property.

$4 (21 x^{6}) + 4 (- 354 x^{5}) + 4 (2090 x^{4}) + 4 (- 4456 x^{3}) + 4 (- 1035 x^{2}) + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15

Simplify.

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Step 8.15.1

Multiply $21$ by $4$ .

$84 x^{6} + 4 (- 354 x^{5}) + 4 (2090 x^{4}) + 4 (- 4456 x^{3}) + 4 (- 1035 x^{2}) + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15.2

Multiply $- 354$ by $4$ .

$84 x^{6} - 1416 x^{5} + 4 (2090 x^{4}) + 4 (- 4456 x^{3}) + 4 (- 1035 x^{2}) + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15.3

Multiply $2090$ by $4$ .

$84 x^{6} - 1416 x^{5} + 8360 x^{4} + 4 (- 4456 x^{3}) + 4 (- 1035 x^{2}) + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15.4

Multiply $- 4456$ by $4$ .

$84 x^{6} - 1416 x^{5} + 8360 x^{4} - 17824 x^{3} + 4 (- 1035 x^{2}) + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15.5

Multiply $- 1035$ by $4$ .

$84 x^{6} - 1416 x^{5} + 8360 x^{4} - 17824 x^{3} - 4140 x^{2} + 4 (12050 x) + 4 \cdot - 5500$

Step 8.15.6

Multiply $12050$ by $4$ .

$84 x^{6} - 1416 x^{5} + 8360 x^{4} - 17824 x^{3} - 4140 x^{2} + 48200 x + 4 \cdot - 5500$

Step 8.15.7

Multiply $4$ by $- 5500$ .

$84 x^{6} - 1416 x^{5} + 8360 x^{4} - 17824 x^{3} - 4140 x^{2} + 48200 x - 22000$