Calculus Examples

$f (x) = \frac{(3 x^{2} + 5 x - 7) (x^{2} - 3)}{3 x + 5}$

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 $f (x) = (3 x^{2} + 5 x - 7) (x^{2} - 3)$ and $g (x) = 3 x + 5$ .

$\frac{(3 x + 5) \frac{d}{d x} [(3 x^{2} + 5 x - 7) (x^{2} - 3)] - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

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 $f (x) = 3 x^{2} + 5 x - 7$ and $g (x) = x^{2} - 3$ .

$\frac{(3 x + 5) ((3 x^{2} + 5 x - 7) \frac{d}{d x} [x^{2} - 3] + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{(3 x + 5) ((3 x^{2} + 5 x - 7) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]) + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.2

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

$\frac{(3 x + 5) ((3 x^{2} + 5 x - 7) (2 x + \frac{d}{d x} [- 3]) + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.3

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

$\frac{(3 x + 5) ((3 x^{2} + 5 x - 7) (2 x + 0) + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $2 x$ and $0$ .

$\frac{(3 x + 5) ((3 x^{2} + 5 x - 7) (2 x) + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.4.2

Move $2$ to the left of $3 x^{2} + 5 x - 7$ .

$\frac{(3 x + 5) (2 \cdot (3 x^{2} + 5 x - 7) x + (x^{2} - 3) \frac{d}{d x} [3 x^{2} + 5 x - 7]) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.5

By the Sum Rule, the derivative of $3 x^{2} + 5 x - 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.6

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.7

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

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (3 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.8

Multiply $2$ by $3$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.9

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.10

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

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + 5 \cdot 1 + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.11

Multiply $5$ by $1$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + 5 + \frac{d}{d x} [- 7])) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.12

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + 5 + 0)) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 3.13

Add $6 x + 5$ and $0$ .

$\frac{(3 x + 5) (2 (3 x^{2} + 5 x - 7) x + (x^{2} - 3) (6 x + 5)) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{(3 x + 5) ((2 (3 x^{2}) + 2 (5 x) + 2 \cdot - 7) x + (x^{2} - 3) (6 x + 5)) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{(3 x + 5) (2 (3 x^{2}) x + 2 (5 x) x + 2 \cdot - 7 x + (x^{2} - 3) (6 x + 5)) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.3

Apply the distributive property.

$\frac{(3 x + 5) (2 (3 x^{2}) x + 2 (5 x) x + 2 \cdot - 7 x + (x^{2} - 3) (6 x) + (x^{2} - 3) \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.4

Apply the distributive property.

$\frac{(3 x + 5) (2 (3 x^{2}) x + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + (x^{2} - 3) \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.5

Apply the distributive property.

$\frac{(3 x + 5) (2 (3 x^{2}) x + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6

Combine terms.

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Step 4.6.1

Multiply $3$ by $2$ .

$\frac{(3 x + 5) (6 x^{2} x + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.2

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

$\frac{(3 x + 5) (6 (x^{1} x^{2}) + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.3

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

$\frac{(3 x + 5) (6 x^{1 + 2} + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.4

Add $1$ and $2$ .

$\frac{(3 x + 5) (6 x^{3} + 2 (5 x) x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.5

Multiply $5$ by $2$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x \cdot x + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.6

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

$\frac{(3 x + 5) (6 x^{3} + 10 (x^{1} x) + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.7

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

$\frac{(3 x + 5) (6 x^{3} + 10 (x^{1} x^{1}) + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.8

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

$\frac{(3 x + 5) (6 x^{3} + 10 x^{1 + 1} + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.9

Add $1$ and $1$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} + 2 \cdot - 7 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.10

Multiply $2$ by $- 7$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + x^{2} (6 x) - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.11

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

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Step 4.6.11.1

Move $x$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + x \cdot x^{2} \cdot 6 - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.11.2

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

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Step 4.6.11.2.1

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

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + x^{1} x^{2} \cdot 6 - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.11.2.2

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

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + x^{1 + 2} \cdot 6 - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.11.3

Add $1$ and $2$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + x^{3} \cdot 6 - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.12

Move $6$ to the left of $x^{3}$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + 6 \cdot x^{3} - 3 (6 x) + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.13

Multiply $6$ by $- 3$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + 6 x^{3} - 18 x + x^{2} \cdot 5 - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.14

Move $5$ to the left of $x^{2}$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + 6 x^{3} - 18 x + 5 \cdot x^{2} - 3 \cdot 5) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.15

Multiply $- 3$ by $5$ .

$\frac{(3 x + 5) (6 x^{3} + 10 x^{2} - 14 x + 6 x^{3} - 18 x + 5 x^{2} - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.16

Add $6 x^{3}$ and $6 x^{3}$ .

$\frac{(3 x + 5) (12 x^{3} + 10 x^{2} - 14 x - 18 x + 5 x^{2} - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.17

Subtract $18 x$ from $- 14 x$ .

$\frac{(3 x + 5) (12 x^{3} + 10 x^{2} - 32 x + 5 x^{2} - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 4.6.18

Add $10 x^{2}$ and $5 x^{2}$ .

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \frac{d}{d x} [3 x + 5]}{{(3 x + 5)}^{2}}$

Step 5

By the Sum Rule, the derivative of $3 x + 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [5]$ .

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) (\frac{d}{d x} [3 x] + \frac{d}{d x} [5])}{{(3 x + 5)}^{2}}$

Step 6

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 6.1

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

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) (3 \frac{d}{d x} [x] + \frac{d}{d x} [5])}{{(3 x + 5)}^{2}}$

Step 6.2

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

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) (3 \cdot 1 + \frac{d}{d x} [5])}{{(3 x + 5)}^{2}}$

Step 6.3

Multiply $3$ by $1$ .

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) (3 + \frac{d}{d x} [5])}{{(3 x + 5)}^{2}}$

Step 7

Differentiate using the Constant Rule.

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Step 7.1

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

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) (3 + 0)}{{(3 x + 5)}^{2}}$

Step 7.2

Add $3$ and $0$ .

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) - (3 x^{2} + 5 x - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15) + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Expand $(3 x + 5) (12 x^{3} + 15 x^{2} - 32 x - 15)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{3 x (12 x^{3}) + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2

Simplify each term.

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Step 8.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot 12 x \cdot x^{3} + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.2

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

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Step 8.2.1.2.2.1

Move $x^{3}$ .

$\frac{3 \cdot 12 (x^{3} x) + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.2.2

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

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Step 8.2.1.2.2.2.1

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

$\frac{3 \cdot 12 (x^{3} x^{1}) + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.2.2.2

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

$\frac{3 \cdot 12 x^{3 + 1} + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.2.3

Add $3$ and $1$ .

$\frac{3 \cdot 12 x^{4} + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.3

Multiply $3$ by $12$ .

$\frac{36 x^{4} + 3 x (15 x^{2}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.4

Rewrite using the commutative property of multiplication.

$\frac{36 x^{4} + 3 \cdot 15 x \cdot x^{2} + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.5

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

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Step 8.2.1.2.5.1

Move $x^{2}$ .

$\frac{36 x^{4} + 3 \cdot 15 (x^{2} x) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.5.2

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

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Step 8.2.1.2.5.2.1

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

$\frac{36 x^{4} + 3 \cdot 15 (x^{2} x^{1}) + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.5.2.2

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

$\frac{36 x^{4} + 3 \cdot 15 x^{2 + 1} + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.5.3

Add $2$ and $1$ .

$\frac{36 x^{4} + 3 \cdot 15 x^{3} + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.6

Multiply $3$ by $15$ .

$\frac{36 x^{4} + 45 x^{3} + 3 x (- 32 x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.7

Rewrite using the commutative property of multiplication.

$\frac{36 x^{4} + 45 x^{3} + 3 \cdot - 32 x \cdot x + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.8

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

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Step 8.2.1.2.8.1

Move $x$ .

$\frac{36 x^{4} + 45 x^{3} + 3 \cdot - 32 (x \cdot x) + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.8.2

Multiply $x$ by $x$ .

$\frac{36 x^{4} + 45 x^{3} + 3 \cdot - 32 x^{2} + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.9

Multiply $3$ by $- 32$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} + 3 x \cdot - 15 + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.10

Multiply $- 15$ by $3$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} - 45 x + 5 (12 x^{3}) + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.11

Multiply $12$ by $5$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} - 45 x + 60 x^{3} + 5 (15 x^{2}) + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.12

Multiply $15$ by $5$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} - 45 x + 60 x^{3} + 75 x^{2} + 5 (- 32 x) + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.13

Multiply $- 32$ by $5$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} - 45 x + 60 x^{3} + 75 x^{2} - 160 x + 5 \cdot - 15 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.2.14

Multiply $5$ by $- 15$ .

$\frac{36 x^{4} + 45 x^{3} - 96 x^{2} - 45 x + 60 x^{3} + 75 x^{2} - 160 x - 75 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.3

Add $45 x^{3}$ and $60 x^{3}$ .

$\frac{36 x^{4} + 105 x^{3} - 96 x^{2} - 45 x + 75 x^{2} - 160 x - 75 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.4

Add $- 96 x^{2}$ and $75 x^{2}$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 45 x - 160 x - 75 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.5

Subtract $160 x$ from $- 45 x$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- (3 x^{2}) - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.6

Simplify each term.

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Step 8.2.1.6.1

Multiply $3$ by $- 1$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{2} - (5 x) - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.6.2

Multiply $5$ by $- 1$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{2} - 5 x - - 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.6.3

Multiply $- 1$ by $- 7$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{2} - 5 x + 7) (x^{2} - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.7

Expand $(- 3 x^{2} - 5 x + 7) (x^{2} - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{2} x^{2} - 3 x^{2} \cdot - 3 - 5 x \cdot x^{2} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8

Simplify each term.

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Step 8.2.1.8.1

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

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Step 8.2.1.8.1.1

Move $x^{2}$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 (x^{2} x^{2}) - 3 x^{2} \cdot - 3 - 5 x \cdot x^{2} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.1.2

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

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{2 + 2} - 3 x^{2} \cdot - 3 - 5 x \cdot x^{2} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.1.3

Add $2$ and $2$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} - 3 x^{2} \cdot - 3 - 5 x \cdot x^{2} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.2

Multiply $- 3$ by $- 3$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 x \cdot x^{2} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.3

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

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Step 8.2.1.8.3.1

Move $x^{2}$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 (x^{2} x) - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.3.2

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

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Step 8.2.1.8.3.2.1

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

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 (x^{2} x^{1}) - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.3.2.2

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

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 x^{2 + 1} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.3.3

Add $2$ and $1$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 x^{3} - 5 x \cdot - 3 + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.4

Multiply $- 3$ by $- 5$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 x^{3} + 15 x + 7 x^{2} + 7 \cdot - 3) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.8.5

Multiply $7$ by $- 3$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 9 x^{2} - 5 x^{3} + 15 x + 7 x^{2} - 21) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.9

Add $9 x^{2}$ and $7 x^{2}$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + (- 3 x^{4} + 16 x^{2} - 5 x^{3} + 15 x - 21) \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.10

Apply the distributive property.

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 3 x^{4} \cdot 3 + 16 x^{2} \cdot 3 - 5 x^{3} \cdot 3 + 15 x \cdot 3 - 21 \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.11

Simplify.

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Step 8.2.1.11.1

Multiply $3$ by $- 3$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 9 x^{4} + 16 x^{2} \cdot 3 - 5 x^{3} \cdot 3 + 15 x \cdot 3 - 21 \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.11.2

Multiply $3$ by $16$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 9 x^{4} + 48 x^{2} - 5 x^{3} \cdot 3 + 15 x \cdot 3 - 21 \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.11.3

Multiply $3$ by $- 5$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 9 x^{4} + 48 x^{2} - 15 x^{3} + 15 x \cdot 3 - 21 \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.11.4

Multiply $3$ by $15$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 9 x^{4} + 48 x^{2} - 15 x^{3} + 45 x - 21 \cdot 3}{{(3 x + 5)}^{2}}$

Step 8.2.1.11.5

Multiply $- 21$ by $3$ .

$\frac{36 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 - 9 x^{4} + 48 x^{2} - 15 x^{3} + 45 x - 63}{{(3 x + 5)}^{2}}$

Step 8.2.2

Subtract $9 x^{4}$ from $36 x^{4}$ .

$\frac{27 x^{4} + 105 x^{3} - 21 x^{2} - 205 x - 75 + 48 x^{2} - 15 x^{3} + 45 x - 63}{{(3 x + 5)}^{2}}$

Step 8.2.3

Subtract $15 x^{3}$ from $105 x^{3}$ .

$\frac{27 x^{4} + 90 x^{3} - 21 x^{2} - 205 x - 75 + 48 x^{2} + 45 x - 63}{{(3 x + 5)}^{2}}$

Step 8.2.4

Add $- 21 x^{2}$ and $48 x^{2}$ .

$\frac{27 x^{4} + 90 x^{3} + 27 x^{2} - 205 x - 75 + 45 x - 63}{{(3 x + 5)}^{2}}$

Step 8.2.5

Add $- 205 x$ and $45 x$ .

$\frac{27 x^{4} + 90 x^{3} + 27 x^{2} - 160 x - 75 - 63}{{(3 x + 5)}^{2}}$

Step 8.2.6

Subtract $63$ from $- 75$ .

$\frac{27 x^{4} + 90 x^{3} + 27 x^{2} - 160 x - 138}{{(3 x + 5)}^{2}}$