Calculus Examples

$\frac{{(x + 2)}^{5} - 32}{x}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(x + 2)}^{5} - 32$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [{(x + 2)}^{5} - 32] - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 2

By the Sum Rule, the derivative of ${(x + 2)}^{5} - 32$ with respect to $x$ is $\frac{d}{d x} [{(x + 2)}^{5}] + \frac{d}{d x} [- 32]$ .

$\frac{x (\frac{d}{d x} [{(x + 2)}^{5}] + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x + 2$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x + 2$ .

$\frac{x (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x + 2] + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$\frac{x (5 u^{4} \frac{d}{d x} [x + 2] + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Replace all occurrences of $u$ with $x + 2$ .

$\frac{x (5 {(x + 2)}^{4} \frac{d}{d x} [x + 2] + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{x (5 {(x + 2)}^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

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

$\frac{x (5 {(x + 2)}^{4} (1 + \frac{d}{d x} [2]) + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{x (5 {(x + 2)}^{4} (1 + 0) + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

$\frac{x (5 {(x + 2)}^{4} \cdot 1 + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.4.2

Multiply $5$ by $1$ .

$\frac{x (5 {(x + 2)}^{4} + \frac{d}{d x} [- 32]) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.5

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32$ with respect to $x$ is $0$ .

$\frac{x (5 {(x + 2)}^{4} + 0) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.6

Add $5 {(x + 2)}^{4}$ and $0$ .

$\frac{x (5 {(x + 2)}^{4}) - ({(x + 2)}^{5} - 32) \frac{d}{d x} [x]}{x^{2}}$

Step 4.7

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

$\frac{x (5 {(x + 2)}^{4}) - ({(x + 2)}^{5} - 32) \cdot 1}{x^{2}}$

Step 4.8

Multiply $- 1$ by $1$ .

$\frac{x (5 {(x + 2)}^{4}) - ({(x + 2)}^{5} - 32)}{x^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{x (5 {(x + 2)}^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite using the commutative property of multiplication.

$\frac{5 x {(x + 2)}^{4} - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.2

Use the Binomial Theorem.

$\frac{5 x (x^{4} + 4 x^{3} \cdot 2 + 6 x^{2} \cdot 2^{2} + 4 x \cdot 2^{3} + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply $2$ by $4$ .

$\frac{5 x (x^{4} + 8 x^{3} + 6 x^{2} \cdot 2^{2} + 4 x \cdot 2^{3} + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3.2

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

$\frac{5 x (x^{4} + 8 x^{3} + 6 x^{2} \cdot 4 + 4 x \cdot 2^{3} + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3.3

Multiply $4$ by $6$ .

$\frac{5 x (x^{4} + 8 x^{3} + 24 x^{2} + 4 x \cdot 2^{3} + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3.4

Raise $2$ to the power of $3$ .

$\frac{5 x (x^{4} + 8 x^{3} + 24 x^{2} + 4 x \cdot 8 + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3.5

Multiply $8$ by $4$ .

$\frac{5 x (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 2^{4}) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.3.6

Raise $2$ to the power of $4$ .

$\frac{5 x (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.4

Apply the distributive property.

$\frac{5 x \cdot x^{4} + 5 x (8 x^{3}) + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5

Simplify.

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Step 5.2.5.1

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

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Step 5.2.5.1.1

Move $x^{4}$ .

$\frac{5 (x^{4} x) + 5 x (8 x^{3}) + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.1.2

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

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Step 5.2.5.1.2.1

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

$\frac{5 (x^{4} x^{1}) + 5 x (8 x^{3}) + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.1.2.2

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

$\frac{5 x^{4 + 1} + 5 x (8 x^{3}) + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.1.3

Add $4$ and $1$ .

$\frac{5 x^{5} + 5 x (8 x^{3}) + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.2

Rewrite using the commutative property of multiplication.

$\frac{5 x^{5} + 5 \cdot 8 x \cdot x^{3} + 5 x (24 x^{2}) + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.3

Rewrite using the commutative property of multiplication.

$\frac{5 x^{5} + 5 \cdot 8 x \cdot x^{3} + 5 \cdot 24 x \cdot x^{2} + 5 x (32 x) + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.4

Rewrite using the commutative property of multiplication.

$\frac{5 x^{5} + 5 \cdot 8 x \cdot x^{3} + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 5 x \cdot 16 - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.5.5

Multiply $16$ by $5$ .

$\frac{5 x^{5} + 5 \cdot 8 x \cdot x^{3} + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6

Simplify each term.

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Step 5.2.6.1

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

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Step 5.2.6.1.1

Move $x^{3}$ .

$\frac{5 x^{5} + 5 \cdot 8 (x^{3} x) + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.1.2

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

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Step 5.2.6.1.2.1

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

$\frac{5 x^{5} + 5 \cdot 8 (x^{3} x^{1}) + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.1.2.2

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

$\frac{5 x^{5} + 5 \cdot 8 x^{3 + 1} + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.1.3

Add $3$ and $1$ .

$\frac{5 x^{5} + 5 \cdot 8 x^{4} + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.2

Multiply $5$ by $8$ .

$\frac{5 x^{5} + 40 x^{4} + 5 \cdot 24 x \cdot x^{2} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.3

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

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Step 5.2.6.3.1

Move $x^{2}$ .

$\frac{5 x^{5} + 40 x^{4} + 5 \cdot 24 (x^{2} x) + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.3.2

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

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Step 5.2.6.3.2.1

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

$\frac{5 x^{5} + 40 x^{4} + 5 \cdot 24 (x^{2} x^{1}) + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.3.2.2

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

$\frac{5 x^{5} + 40 x^{4} + 5 \cdot 24 x^{2 + 1} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.3.3

Add $2$ and $1$ .

$\frac{5 x^{5} + 40 x^{4} + 5 \cdot 24 x^{3} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.4

Multiply $5$ by $24$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 5 \cdot 32 x \cdot x + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.5

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

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Step 5.2.6.5.1

Move $x$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 5 \cdot 32 (x \cdot x) + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.5.2

Multiply $x$ by $x$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 5 \cdot 32 x^{2} + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.6.6

Multiply $5$ by $32$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - {(x + 2)}^{5} - - 32}{x^{2}}$

Step 5.2.7

Use the Binomial Theorem.

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 5 x^{4} \cdot 2 + 10 x^{3} \cdot 2^{2} + 10 x^{2} \cdot 2^{3} + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8

Simplify each term.

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Step 5.2.8.1

Multiply $2$ by $5$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 10 x^{3} \cdot 2^{2} + 10 x^{2} \cdot 2^{3} + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.2

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

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 10 x^{3} \cdot 4 + 10 x^{2} \cdot 2^{3} + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.3

Multiply $4$ by $10$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 10 x^{2} \cdot 2^{3} + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.4

Raise $2$ to the power of $3$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 10 x^{2} \cdot 8 + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.5

Multiply $8$ by $10$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 5 x \cdot 2^{4} + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.6

Raise $2$ to the power of $4$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 5 x \cdot 16 + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.7

Multiply $16$ by $5$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 2^{5}) - - 32}{x^{2}}$

Step 5.2.8.8

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

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - (x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 32) - - 32}{x^{2}}$

Step 5.2.9

Apply the distributive property.

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - (10 x^{4}) - (40 x^{3}) - (80 x^{2}) - (80 x) - 1 \cdot 32 - - 32}{x^{2}}$

Step 5.2.10

Simplify.

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Step 5.2.10.1

Multiply $10$ by $- 1$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - (40 x^{3}) - (80 x^{2}) - (80 x) - 1 \cdot 32 - - 32}{x^{2}}$

Step 5.2.10.2

Multiply $40$ by $- 1$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - 40 x^{3} - (80 x^{2}) - (80 x) - 1 \cdot 32 - - 32}{x^{2}}$

Step 5.2.10.3

Multiply $80$ by $- 1$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - 40 x^{3} - 80 x^{2} - (80 x) - 1 \cdot 32 - - 32}{x^{2}}$

Step 5.2.10.4

Multiply $80$ by $- 1$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - 40 x^{3} - 80 x^{2} - 80 x - 1 \cdot 32 - - 32}{x^{2}}$

Step 5.2.10.5

Multiply $- 1$ by $32$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - 40 x^{3} - 80 x^{2} - 80 x - 32 - - 32}{x^{2}}$

Step 5.2.11

Multiply $- 1$ by $- 32$ .

$\frac{5 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - x^{5} - 10 x^{4} - 40 x^{3} - 80 x^{2} - 80 x - 32 + 32}{x^{2}}$

Step 5.2.12

Subtract $x^{5}$ from $5 x^{5}$ .

$\frac{4 x^{5} + 40 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - 10 x^{4} - 40 x^{3} - 80 x^{2} - 80 x - 32 + 32}{x^{2}}$

Step 5.2.13

Subtract $10 x^{4}$ from $40 x^{4}$ .

$\frac{4 x^{5} + 30 x^{4} + 120 x^{3} + 160 x^{2} + 80 x - 40 x^{3} - 80 x^{2} - 80 x - 32 + 32}{x^{2}}$

Step 5.2.14

Subtract $40 x^{3}$ from $120 x^{3}$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 160 x^{2} + 80 x - 80 x^{2} - 80 x - 32 + 32}{x^{2}}$

Step 5.2.15

Subtract $80 x^{2}$ from $160 x^{2}$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2} + 80 x - 80 x - 32 + 32}{x^{2}}$

Step 5.2.16

Subtract $80 x$ from $80 x$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2} + 0 - 32 + 32}{x^{2}}$

Step 5.2.17

Add $4 x^{5}$ and $0$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2} - 32 + 32}{x^{2}}$

Step 5.2.18

Add $- 32$ and $32$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2} + 0}{x^{2}}$

Step 5.2.19

Add $4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2}$ and $0$ .

$\frac{4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2}}{x^{2}}$

Step 5.2.20

Factor $2 x^{2}$ out of $4 x^{5} + 30 x^{4} + 80 x^{3} + 80 x^{2}$ .

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Step 5.2.20.1

Factor $2 x^{2}$ out of $4 x^{5}$ .

$\frac{2 x^{2} (2 x^{3}) + 30 x^{4} + 80 x^{3} + 80 x^{2}}{x^{2}}$

Step 5.2.20.2

Factor $2 x^{2}$ out of $30 x^{4}$ .

$\frac{2 x^{2} (2 x^{3}) + 2 x^{2} (15 x^{2}) + 80 x^{3} + 80 x^{2}}{x^{2}}$

Step 5.2.20.3

Factor $2 x^{2}$ out of $80 x^{3}$ .

$\frac{2 x^{2} (2 x^{3}) + 2 x^{2} (15 x^{2}) + 2 x^{2} (40 x) + 80 x^{2}}{x^{2}}$

Step 5.2.20.4

Factor $2 x^{2}$ out of $80 x^{2}$ .

$\frac{2 x^{2} (2 x^{3}) + 2 x^{2} (15 x^{2}) + 2 x^{2} (40 x) + 2 x^{2} (40)}{x^{2}}$

Step 5.2.20.5

Factor $2 x^{2}$ out of $2 x^{2} (2 x^{3}) + 2 x^{2} (15 x^{2})$ .

$\frac{2 x^{2} (2 x^{3} + 15 x^{2}) + 2 x^{2} (40 x) + 2 x^{2} (40)}{x^{2}}$

Step 5.2.20.6

Factor $2 x^{2}$ out of $2 x^{2} (2 x^{3} + 15 x^{2}) + 2 x^{2} (40 x)$ .

$\frac{2 x^{2} (2 x^{3} + 15 x^{2} + 40 x) + 2 x^{2} (40)}{x^{2}}$

Step 5.2.20.7

Factor $2 x^{2}$ out of $2 x^{2} (2 x^{3} + 15 x^{2} + 40 x) + 2 x^{2} (40)$ .

$\frac{2 x^{2} (2 x^{3} + 15 x^{2} + 40 x + 40)}{x^{2}}$

Step 5.3

Cancel the common factor of $x^{2}$ .

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Step 5.3.1

Cancel the common factor.

$\frac{2 x^{2} (2 x^{3} + 15 x^{2} + 40 x + 40)}{x^{2}}$

Step 5.3.2

Divide $2 (2 x^{3} + 15 x^{2} + 40 x + 40)$ by $1$ .

$2 (2 x^{3} + 15 x^{2} + 40 x + 40)$

Step 5.4

Apply the distributive property.

$2 (2 x^{3}) + 2 (15 x^{2}) + 2 (40 x) + 2 \cdot 40$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $2$ by $2$ .

$4 x^{3} + 2 (15 x^{2}) + 2 (40 x) + 2 \cdot 40$

Step 5.5.2

Multiply $15$ by $2$ .

$4 x^{3} + 30 x^{2} + 2 (40 x) + 2 \cdot 40$

Step 5.5.3

Multiply $40$ by $2$ .

$4 x^{3} + 30 x^{2} + 80 x + 2 \cdot 40$

Step 5.5.4

Multiply $2$ by $40$ .

$4 x^{3} + 30 x^{2} + 80 x + 80$