Calculus Examples

$f (t) = 10 (\frac{2 t^{2} + 8 t + 55}{t^{2} + 4 t + 10})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $10$ and $\frac{2 t^{2} + 8 t + 55}{t^{2} + 4 t + 10}$ .

$\frac{d}{d t} [\frac{10 (2 t^{2} + 8 t + 55)}{t^{2} + 4 t + 10}]$

Step 1.2

Since $10$ is constant with respect to $t$ , the derivative of $\frac{10 (2 t^{2} + 8 t + 55)}{t^{2} + 4 t + 10}$ with respect to $t$ is $10 \frac{d}{d t} [\frac{2 t^{2} + 8 t + 55}{t^{2} + 4 t + 10}]$ .

$10 \frac{d}{d t} [\frac{2 t^{2} + 8 t + 55}{t^{2} + 4 t + 10}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = 2 t^{2} + 8 t + 55$ and $g (t) = t^{2} + 4 t + 10$ .

$10 \frac{(t^{2} + 4 t + 10) \frac{d}{d t} [2 t^{2} + 8 t + 55] - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $2 t^{2} + 8 t + 55$ with respect to $t$ is $\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [8 t] + \frac{d}{d t} [55]$ .

$10 \frac{(t^{2} + 4 t + 10) (\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [8 t] + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.2

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

$10 \frac{(t^{2} + 4 t + 10) (2 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [8 t] + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.3

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

$10 \frac{(t^{2} + 4 t + 10) (2 (2 t) + \frac{d}{d t} [8 t] + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.4

Multiply $2$ by $2$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + \frac{d}{d t} [8 t] + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.5

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

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8 \frac{d}{d t} [t] + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.6

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

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8 \cdot 1 + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.7

Multiply $8$ by $1$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8 + \frac{d}{d t} [55]) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.8

Since $55$ is constant with respect to $t$ , the derivative of $55$ with respect to $t$ is $0$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8 + 0) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.9

Add $4 t + 8$ and $0$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) \frac{d}{d t} [t^{2} + 4 t + 10]}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.10

By the Sum Rule, the derivative of $t^{2} + 4 t + 10$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [4 t] + \frac{d}{d t} [10]$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [4 t] + \frac{d}{d t} [10])}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.11

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

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + \frac{d}{d t} [4 t] + \frac{d}{d t} [10])}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.12

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

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4 \frac{d}{d t} [t] + \frac{d}{d t} [10])}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.13

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

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4 \cdot 1 + \frac{d}{d t} [10])}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.14

Multiply $4$ by $1$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4 + \frac{d}{d t} [10])}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.15

Since $10$ is constant with respect to $t$ , the derivative of $10$ with respect to $t$ is $0$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4 + 0)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.16

Combine fractions.

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Step 3.16.1

Add $2 t + 4$ and $0$ .

$10 \frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 3.16.2

Combine $10$ and $\frac{(t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4)}{{(t^{2} + 4 t + 10)}^{2}}$ .

$\frac{10 ((t^{2} + 4 t + 10) (4 t + 8) - (2 t^{2} + 8 t + 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{10 ((t^{2} + 4 t + 10) (4 t + 8) + (- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{10 ((t^{2} + 4 t + 10) (4 t + 8)) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Expand $(t^{2} + 4 t + 10) (4 t + 8)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{10 (t^{2} (4 t) + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2

Simplify each term.

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Step 4.3.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{10 (4 t^{2} t + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.2

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

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Step 4.3.1.2.2.1

Move $t$ .

$\frac{10 (4 (t \cdot t^{2}) + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.2.2

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

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Step 4.3.1.2.2.2.1

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

$\frac{10 (4 (t^{1} t^{2}) + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.2.2.2

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

$\frac{10 (4 t^{1 + 2} + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.2.3

Add $1$ and $2$ .

$\frac{10 (4 t^{3} + t^{2} \cdot 8 + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.3

Move $8$ to the left of $t^{2}$ .

$\frac{10 (4 t^{3} + 8 \cdot t^{2} + 4 t (4 t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.4

Rewrite using the commutative property of multiplication.

$\frac{10 (4 t^{3} + 8 t^{2} + 4 \cdot 4 t \cdot t + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.5

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.2.5.1

Move $t$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 4 \cdot 4 (t \cdot t) + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.5.2

Multiply $t$ by $t$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 4 \cdot 4 t^{2} + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.6

Multiply $4$ by $4$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 16 t^{2} + 4 t \cdot 8 + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.7

Multiply $8$ by $4$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 16 t^{2} + 32 t + 10 (4 t) + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.8

Multiply $4$ by $10$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 16 t^{2} + 32 t + 40 t + 10 \cdot 8) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.2.9

Multiply $10$ by $8$ .

$\frac{10 (4 t^{3} + 8 t^{2} + 16 t^{2} + 32 t + 40 t + 80) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.3

Add $8 t^{2}$ and $16 t^{2}$ .

$\frac{10 (4 t^{3} + 24 t^{2} + 32 t + 40 t + 80) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.4

Add $32 t$ and $40 t$ .

$\frac{10 (4 t^{3} + 24 t^{2} + 72 t + 80) + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.5

Apply the distributive property.

$\frac{10 (4 t^{3}) + 10 (24 t^{2}) + 10 (72 t) + 10 \cdot 80 + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.6

Simplify.

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Step 4.3.1.6.1

Multiply $4$ by $10$ .

$\frac{40 t^{3} + 10 (24 t^{2}) + 10 (72 t) + 10 \cdot 80 + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.6.2

Multiply $24$ by $10$ .

$\frac{40 t^{3} + 240 t^{2} + 10 (72 t) + 10 \cdot 80 + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.6.3

Multiply $72$ by $10$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 10 \cdot 80 + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.6.4

Multiply $10$ by $80$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 ((- (2 t^{2}) - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.7

Simplify each term.

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Step 4.3.1.7.1

Multiply $2$ by $- 1$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 ((- 2 t^{2} - (8 t) - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.7.2

Multiply $8$ by $- 1$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 ((- 2 t^{2} - 8 t - 1 \cdot 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.7.3

Multiply $- 1$ by $55$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 ((- 2 t^{2} - 8 t - 55) (2 t + 4))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.8

Expand $(- 2 t^{2} - 8 t - 55) (2 t + 4)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 t^{2} (2 t) - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9

Simplify each term.

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Step 4.3.1.9.1

Rewrite using the commutative property of multiplication.

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 \cdot 2 t^{2} t - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.2

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

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Step 4.3.1.9.2.1

Move $t$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 \cdot 2 (t \cdot t^{2}) - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.2.2

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

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Step 4.3.1.9.2.2.1

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

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 \cdot 2 (t^{1} t^{2}) - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.2.2.2

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

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 \cdot 2 t^{1 + 2} - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.2.3

Add $1$ and $2$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 2 \cdot 2 t^{3} - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.3

Multiply $- 2$ by $2$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 2 t^{2} \cdot 4 - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.4

Multiply $4$ by $- 2$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 8 t (2 t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.5

Rewrite using the commutative property of multiplication.

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 8 \cdot 2 t \cdot t - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.6

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.9.6.1

Move $t$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 8 \cdot 2 (t \cdot t) - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.6.2

Multiply $t$ by $t$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 8 \cdot 2 t^{2} - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.7

Multiply $- 8$ by $2$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 16 t^{2} - 8 t \cdot 4 - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.8

Multiply $4$ by $- 8$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 16 t^{2} - 32 t - 55 (2 t) - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.9

Multiply $2$ by $- 55$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 16 t^{2} - 32 t - 110 t - 55 \cdot 4)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.9.10

Multiply $- 55$ by $4$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 8 t^{2} - 16 t^{2} - 32 t - 110 t - 220)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.10

Subtract $16 t^{2}$ from $- 8 t^{2}$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 24 t^{2} - 32 t - 110 t - 220)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.11

Subtract $110 t$ from $- 32 t$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3} - 24 t^{2} - 142 t - 220)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.12

Apply the distributive property.

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 + 10 (- 4 t^{3}) + 10 (- 24 t^{2}) + 10 (- 142 t) + 10 \cdot - 220}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.13

Simplify.

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Step 4.3.1.13.1

Multiply $- 4$ by $10$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 - 40 t^{3} + 10 (- 24 t^{2}) + 10 (- 142 t) + 10 \cdot - 220}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.13.2

Multiply $- 24$ by $10$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 - 40 t^{3} - 240 t^{2} + 10 (- 142 t) + 10 \cdot - 220}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.13.3

Multiply $- 142$ by $10$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 - 40 t^{3} - 240 t^{2} - 1420 t + 10 \cdot - 220}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.1.13.4

Multiply $10$ by $- 220$ .

$\frac{40 t^{3} + 240 t^{2} + 720 t + 800 - 40 t^{3} - 240 t^{2} - 1420 t - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.2

Combine the opposite terms in $40 t^{3} + 240 t^{2} + 720 t + 800 - 40 t^{3} - 240 t^{2} - 1420 t - 2200$ .

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Step 4.3.2.1

Subtract $40 t^{3}$ from $40 t^{3}$ .

$\frac{240 t^{2} + 720 t + 800 + 0 - 240 t^{2} - 1420 t - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.2.2

Add $240 t^{2} + 720 t + 800$ and $0$ .

$\frac{240 t^{2} + 720 t + 800 - 240 t^{2} - 1420 t - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.2.3

Subtract $240 t^{2}$ from $240 t^{2}$ .

$\frac{720 t + 800 + 0 - 1420 t - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.2.4

Add $720 t + 800$ and $0$ .

$\frac{720 t + 800 - 1420 t - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.3

Subtract $1420 t$ from $720 t$ .

$\frac{- 700 t + 800 - 2200}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.3.4

Subtract $2200$ from $800$ .

$\frac{- 700 t - 1400}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.4

Factor $700$ out of $- 700 t - 1400$ .

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Step 4.4.1

Factor $700$ out of $- 700 t$ .

$\frac{700 (- t) - 1400}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.4.2

Factor $700$ out of $- 1400$ .

$\frac{700 (- t) + 700 (- 2)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.4.3

Factor $700$ out of $700 (- t) + 700 (- 2)$ .

$\frac{700 (- t - 2)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.5

Factor $- 1$ out of $- t$ .

$\frac{700 (- (t) - 2)}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.6

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{700 (- (t) - 1 (2))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.7

Factor $- 1$ out of $- (t) - 1 (2)$ .

$\frac{700 (- (t + 2))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.8

Rewrite $- (t + 2)$ as $- 1 (t + 2)$ .

$\frac{700 (- 1 (t + 2))}{{(t^{2} + 4 t + 10)}^{2}}$

Step 4.9

Move the negative in front of the fraction.

$- \frac{700 (t + 2)}{{(t^{2} + 4 t + 10)}^{2}}$