Calculus Examples

$\frac{2 u^{2}}{{(u^{2} + u)}^{3}}$

Step 1

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

$2 \frac{d}{d u} [\frac{u^{2}}{{(u^{2} + u)}^{3}}]$

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(u^{2} + u)}^{3})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Multiply $3$ by $2$ .

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

Step 3.2

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

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

Step 3.3

Move $2$ to the left of ${(u^{2} + u)}^{3}$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = u^{3}$ and $g (u) = u^{2} + u$ .

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Step 4.1

To apply the Chain Rule, set $u_{1}$ as $u^{2} + u$ .

$2 \frac{2 {(u^{2} + u)}^{3} u - u^{2} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d u} [u^{2} + u])}{{(u^{2} + u)}^{6}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$2 \frac{2 {(u^{2} + u)}^{3} u - u^{2} (3 {u_{1}}^{2} \frac{d}{d u} [u^{2} + u])}{{(u^{2} + u)}^{6}}$

Step 4.3

Replace all occurrences of $u_{1}$ with $u^{2} + u$ .

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

Step 5

Differentiate.

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Step 5.1

Multiply $3$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

$2 \frac{2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 5.5

Combine $2$ and $\frac{2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1))}{{(u^{2} + u)}^{6}}$ .

$\frac{2 (2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{2 (2 {(u^{2} + u)}^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Use the Binomial Theorem.

$\frac{2 (2 ({(u^{2})}^{3} + 3 {(u^{2})}^{2} u + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Multiply the exponents in ${(u^{2})}^{3}$ .

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Step 6.2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 (2 (u^{2 \cdot 3} + 3 {(u^{2})}^{2} u + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.1.2

Multiply $2$ by $3$ .

$\frac{2 (2 (u^{6} + 3 {(u^{2})}^{2} u + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.2

Multiply the exponents in ${(u^{2})}^{2}$ .

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Step 6.2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 (2 (u^{6} + 3 u^{2 \cdot 2} u + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.2.2

Multiply $2$ by $2$ .

$\frac{2 (2 (u^{6} + 3 u^{4} u + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.3

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

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Step 6.2.2.3.1

Move $u$ .

$\frac{2 (2 (u^{6} + 3 (u \cdot u^{4}) + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.3.2

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

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Step 6.2.2.3.2.1

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

$\frac{2 (2 (u^{6} + 3 (u^{1} u^{4}) + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.3.2.2

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

$\frac{2 (2 (u^{6} + 3 u^{1 + 4} + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.3.3

Add $1$ and $4$ .

$\frac{2 (2 (u^{6} + 3 u^{5} + 3 u^{2} u^{2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.4

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

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Step 6.2.2.4.1

Move $u^{2}$ .

$\frac{2 (2 (u^{6} + 3 u^{5} + 3 (u^{2} u^{2}) + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.4.2

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

$\frac{2 (2 (u^{6} + 3 u^{5} + 3 u^{2 + 2} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.2.4.3

Add $2$ and $2$ .

$\frac{2 (2 (u^{6} + 3 u^{5} + 3 u^{4} + u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.3

Apply the distributive property.

$\frac{2 ((2 u^{6} + 2 (3 u^{5}) + 2 (3 u^{4}) + 2 u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.4

Simplify.

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Step 6.2.4.1

Multiply $3$ by $2$ .

$\frac{2 ((2 u^{6} + 6 u^{5} + 2 (3 u^{4}) + 2 u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.4.2

Multiply $3$ by $2$ .

$\frac{2 ((2 u^{6} + 6 u^{5} + 6 u^{4} + 2 u^{3}) u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.5

Apply the distributive property.

$\frac{2 (2 u^{6} u + 6 u^{5} u + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6

Simplify.

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Step 6.2.6.1

Multiply $u^{6}$ by $u$ by adding the exponents.

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Step 6.2.6.1.1

Move $u$ .

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

Step 6.2.6.1.2

Multiply $u$ by $u^{6}$ .

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Step 6.2.6.1.2.1

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

$\frac{2 (2 (u^{1} u^{6}) + 6 u^{5} u + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.1.2.2

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

$\frac{2 (2 u^{1 + 6} + 6 u^{5} u + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.1.3

Add $1$ and $6$ .

$\frac{2 (2 u^{7} + 6 u^{5} u + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.2

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

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Step 6.2.6.2.1

Move $u$ .

$\frac{2 (2 u^{7} + 6 (u \cdot u^{5}) + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.2.2

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

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Step 6.2.6.2.2.1

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

$\frac{2 (2 u^{7} + 6 (u^{1} u^{5}) + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.2.2.2

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

$\frac{2 (2 u^{7} + 6 u^{1 + 5} + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.2.3

Add $1$ and $5$ .

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{4} u + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.3

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

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Step 6.2.6.3.1

Move $u$ .

$\frac{2 (2 u^{7} + 6 u^{6} + 6 (u \cdot u^{4}) + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.3.2

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

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Step 6.2.6.3.2.1

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

$\frac{2 (2 u^{7} + 6 u^{6} + 6 (u^{1} u^{4}) + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.3.2.2

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

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{1 + 4} + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.3.3

Add $1$ and $4$ .

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{5} + 2 u^{3} u) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.4

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

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Step 6.2.6.4.1

Move $u$ .

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{5} + 2 (u \cdot u^{3})) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.4.2

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

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Step 6.2.6.4.2.1

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

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{5} + 2 (u^{1} u^{3})) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.4.2.2

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

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{5} + 2 u^{1 + 3}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.6.4.3

Add $1$ and $3$ .

$\frac{2 (2 u^{7} + 6 u^{6} + 6 u^{5} + 2 u^{4}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.7

Apply the distributive property.

$\frac{2 (2 u^{7}) + 2 (6 u^{6}) + 2 (6 u^{5}) + 2 (2 u^{4}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.8

Simplify.

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Step 6.2.8.1

Multiply $2$ by $2$ .

$\frac{4 u^{7} + 2 (6 u^{6}) + 2 (6 u^{5}) + 2 (2 u^{4}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.8.2

Multiply $6$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 2 (6 u^{5}) + 2 (2 u^{4}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.8.3

Multiply $6$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 2 (2 u^{4}) + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.8.4

Multiply $2$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.9

Rewrite ${(u^{2} + u)}^{2}$ as $(u^{2} + u) (u^{2} + u)$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{2} + u) (u^{2} + u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.10

Expand $(u^{2} + u) (u^{2} + u)$ using the FOIL Method.

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Step 6.2.10.1

Apply the distributive property.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{2} (u^{2} + u) + u (u^{2} + u)) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.10.2

Apply the distributive property.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{2} u^{2} + u^{2} u + u (u^{2} + u)) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.10.3

Apply the distributive property.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{2} u^{2} + u^{2} u + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11

Simplify and combine like terms.

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Step 6.2.11.1

Simplify each term.

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Step 6.2.11.1.1

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

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Step 6.2.11.1.1.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{2 + 2} + u^{2} u + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.1.2

Add $2$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{2} u + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.2

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

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Step 6.2.11.1.2.1

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

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Step 6.2.11.1.2.1.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{2} u^{1} + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.2.1.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{2 + 1} + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.2.2

Add $2$ and $1$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{3} + u \cdot u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.3

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

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Step 6.2.11.1.3.1

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

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Step 6.2.11.1.3.1.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{3} + u^{1} u^{2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.3.1.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{3} + u^{1 + 2} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.3.2

Add $1$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{3} + u^{3} + u \cdot u) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.1.4

Multiply $u$ by $u$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + u^{3} + u^{3} + u^{2}) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.11.2

Add $u^{3}$ and $u^{3}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} ((u^{4} + 2 u^{3} + u^{2}) (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.12

Expand $(u^{4} + 2 u^{3} + u^{2}) (2 u + 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (u^{4} (2 u) + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13

Simplify each term.

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Step 6.2.13.1

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{4} u + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.2

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

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Step 6.2.13.2.1

Move $u$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 (u \cdot u^{4}) + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.2.2

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

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Step 6.2.13.2.2.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 (u^{1} u^{4}) + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.2.2.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{1 + 4} + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.2.3

Add $1$ and $4$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} \cdot 1 + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.3

Multiply $u^{4}$ by $1$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 u^{3} (2 u) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.4

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 \cdot 2 u^{3} u + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.5

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

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Step 6.2.13.5.1

Move $u$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 \cdot 2 (u \cdot u^{3}) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.5.2

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

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Step 6.2.13.5.2.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 \cdot 2 (u^{1} u^{3}) + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.5.2.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 \cdot 2 u^{1 + 3} + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.5.3

Add $1$ and $3$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 2 \cdot 2 u^{4} + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.6

Multiply $2$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} \cdot 1 + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.7

Multiply $2$ by $1$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + u^{2} (2 u) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.8

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 u^{2} u + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.9

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

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Step 6.2.13.9.1

Move $u$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 (u \cdot u^{2}) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.9.2

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

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Step 6.2.13.9.2.1

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 (u^{1} u^{2}) + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.9.2.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 u^{1 + 2} + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.9.3

Add $1$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 u^{3} + u^{2} \cdot 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.13.10

Multiply $u^{2}$ by $1$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + u^{4} + 4 u^{4} + 2 u^{3} + 2 u^{3} + u^{2}))}{{(u^{2} + u)}^{6}}$

Step 6.2.14

Add $u^{4}$ and $4 u^{4}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + 5 u^{4} + 2 u^{3} + 2 u^{3} + u^{2}))}{{(u^{2} + u)}^{6}}$

Step 6.2.15

Add $2 u^{3}$ and $2 u^{3}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5} + 5 u^{4} + 4 u^{3} + u^{2}))}{{(u^{2} + u)}^{6}}$

Step 6.2.16

Apply the distributive property.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 u^{2} (2 u^{5}) - 3 u^{2} (5 u^{4}) - 3 u^{2} (4 u^{3}) - 3 u^{2} u^{2})}{{(u^{2} + u)}^{6}}$

Step 6.2.17

Simplify.

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Step 6.2.17.1

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 u^{2} (5 u^{4}) - 3 u^{2} (4 u^{3}) - 3 u^{2} u^{2})}{{(u^{2} + u)}^{6}}$

Step 6.2.17.2

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 \cdot 5 u^{2} u^{4} - 3 u^{2} (4 u^{3}) - 3 u^{2} u^{2})}{{(u^{2} + u)}^{6}}$

Step 6.2.17.3

Rewrite using the commutative property of multiplication.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{2} u^{2})}{{(u^{2} + u)}^{6}}$

Step 6.2.17.4

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

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Step 6.2.17.4.1

Move $u^{2}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 (u^{2} u^{2}))}{{(u^{2} + u)}^{6}}$

Step 6.2.17.4.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{2 + 2})}{{(u^{2} + u)}^{6}}$

Step 6.2.17.4.3

Add $2$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{2} u^{5} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18

Simplify each term.

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Step 6.2.18.1

Multiply $u^{2}$ by $u^{5}$ by adding the exponents.

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Step 6.2.18.1.1

Move $u^{5}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 (u^{5} u^{2}) - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.1.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{5 + 2} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.1.3

Add $5$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 3 \cdot 2 u^{7} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.2

Multiply $- 3$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 3 \cdot 5 u^{2} u^{4} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.3

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

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Step 6.2.18.3.1

Move $u^{4}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 3 \cdot 5 (u^{4} u^{2}) - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.3.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 3 \cdot 5 u^{4 + 2} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.3.3

Add $4$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 3 \cdot 5 u^{6} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.4

Multiply $- 3$ by $5$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 15 u^{6} - 3 \cdot 4 u^{2} u^{3} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.5

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

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Step 6.2.18.5.1

Move $u^{3}$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 15 u^{6} - 3 \cdot 4 (u^{3} u^{2}) - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.5.2

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

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 15 u^{6} - 3 \cdot 4 u^{3 + 2} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.5.3

Add $3$ and $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 15 u^{6} - 3 \cdot 4 u^{5} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.18.6

Multiply $- 3$ by $4$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7} - 15 u^{6} - 12 u^{5} - 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.19

Apply the distributive property.

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} + 2 (- 6 u^{7}) + 2 (- 15 u^{6}) + 2 (- 12 u^{5}) + 2 (- 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.20

Simplify.

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Step 6.2.20.1

Multiply $- 6$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} - 12 u^{7} + 2 (- 15 u^{6}) + 2 (- 12 u^{5}) + 2 (- 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.20.2

Multiply $- 15$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} - 12 u^{7} - 30 u^{6} + 2 (- 12 u^{5}) + 2 (- 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.20.3

Multiply $- 12$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} - 12 u^{7} - 30 u^{6} - 24 u^{5} + 2 (- 3 u^{4})}{{(u^{2} + u)}^{6}}$

Step 6.2.20.4

Multiply $- 3$ by $2$ .

$\frac{4 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} - 12 u^{7} - 30 u^{6} - 24 u^{5} - 6 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.21

Subtract $12 u^{7}$ from $4 u^{7}$ .

$\frac{- 8 u^{7} + 12 u^{6} + 12 u^{5} + 4 u^{4} - 30 u^{6} - 24 u^{5} - 6 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.22

Subtract $30 u^{6}$ from $12 u^{6}$ .

$\frac{- 8 u^{7} - 18 u^{6} + 12 u^{5} + 4 u^{4} - 24 u^{5} - 6 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.23

Subtract $24 u^{5}$ from $12 u^{5}$ .

$\frac{- 8 u^{7} - 18 u^{6} - 12 u^{5} + 4 u^{4} - 6 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.24

Subtract $6 u^{4}$ from $4 u^{4}$ .

$\frac{- 8 u^{7} - 18 u^{6} - 12 u^{5} - 2 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.25

Rewrite $- 8 u^{7} - 18 u^{6} - 12 u^{5} - 2 u^{4}$ in a factored form.

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Step 6.2.25.1

Factor $2 u^{4}$ out of $- 8 u^{7} - 18 u^{6} - 12 u^{5} - 2 u^{4}$ .

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Step 6.2.25.1.1

Factor $2 u^{4}$ out of $- 8 u^{7}$ .

$\frac{2 u^{4} (- 4 u^{3}) - 18 u^{6} - 12 u^{5} - 2 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.2

Factor $2 u^{4}$ out of $- 18 u^{6}$ .

$\frac{2 u^{4} (- 4 u^{3}) + 2 u^{4} (- 9 u^{2}) - 12 u^{5} - 2 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.3

Factor $2 u^{4}$ out of $- 12 u^{5}$ .

$\frac{2 u^{4} (- 4 u^{3}) + 2 u^{4} (- 9 u^{2}) + 2 u^{4} (- 6 u) - 2 u^{4}}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.4

Factor $2 u^{4}$ out of $- 2 u^{4}$ .

$\frac{2 u^{4} (- 4 u^{3}) + 2 u^{4} (- 9 u^{2}) + 2 u^{4} (- 6 u) + 2 u^{4} (- 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.5

Factor $2 u^{4}$ out of $2 u^{4} (- 4 u^{3}) + 2 u^{4} (- 9 u^{2})$ .

$\frac{2 u^{4} (- 4 u^{3} - 9 u^{2}) + 2 u^{4} (- 6 u) + 2 u^{4} (- 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.6

Factor $2 u^{4}$ out of $2 u^{4} (- 4 u^{3} - 9 u^{2}) + 2 u^{4} (- 6 u)$ .

$\frac{2 u^{4} (- 4 u^{3} - 9 u^{2} - 6 u) + 2 u^{4} (- 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.1.7

Factor $2 u^{4}$ out of $2 u^{4} (- 4 u^{3} - 9 u^{2} - 6 u) + 2 u^{4} (- 1)$ .

$\frac{2 u^{4} (- 4 u^{3} - 9 u^{2} - 6 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.2

Factor $- 4 u^{3} - 9 u^{2} - 6 u - 1$ using the rational roots test.

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Step 6.2.25.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 4, \pm 2$

Step 6.2.25.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.25, \pm 0.5$

Step 6.2.25.2.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 6.2.25.2.3.1

Substitute $- 1$ into the polynomial.

$- 4 {(- 1)}^{3} - 9 {(- 1)}^{2} - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.2

Raise $- 1$ to the power of $3$ .

$- 4 \cdot - 1 - 9 {(- 1)}^{2} - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.3

Multiply $- 4$ by $- 1$ .

$4 - 9 {(- 1)}^{2} - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.4

Raise $- 1$ to the power of $2$ .

$4 - 9 \cdot 1 - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.5

Multiply $- 9$ by $1$ .

$4 - 9 - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.6

Subtract $9$ from $4$ .

$- 5 - 6 \cdot - 1 - 1$

Step 6.2.25.2.3.7

Multiply $- 6$ by $- 1$ .

$- 5 + 6 - 1$

Step 6.2.25.2.3.8

Add $- 5$ and $6$ .

$1 - 1$

Step 6.2.25.2.3.9

Subtract $1$ from $1$ .

$0$

Step 6.2.25.2.4

Since $- 1$ is a known root, divide the polynomial by $u + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 4 u^{3} - 9 u^{2} - 6 u - 1}{u + 1}$

Step 6.2.25.2.5

Divide $- 4 u^{3} - 9 u^{2} - 6 u - 1$ by $u + 1$ .

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Step 6.2.25.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$u$

$1$

$4 u^{3}$

$9 u^{2}$

$6 u$

$1$

Step 6.2.25.2.5.2

Divide the highest order term in the dividend $- 4 u^{3}$ by the highest order term in divisor $u$ .

				-	$4 u^{2}$
	$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$

Step 6.2.25.2.5.3

Multiply the new quotient term by the divisor.

			-	$4 u^{2}$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			-	$4 u^{3}$	-	$4 u^{2}$

Step 6.2.25.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 u^{3} - 4 u^{2}$

			-	$4 u^{2}$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$

Step 6.2.25.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$4 u^{2}$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$

Step 6.2.25.2.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$4 u^{2}$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$

Step 6.2.25.2.5.7

Divide the highest order term in the dividend $- 5 u^{2}$ by the highest order term in divisor $u$ .

			-	$4 u^{2}$	-	$5 u$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$

Step 6.2.25.2.5.8

Multiply the new quotient term by the divisor.

			-	$4 u^{2}$	-	$5 u$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					-	$5 u^{2}$	-	$5 u$

Step 6.2.25.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 u^{2} - 5 u$

			-	$4 u^{2}$	-	$5 u$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$

Step 6.2.25.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$4 u^{2}$	-	$5 u$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$

Step 6.2.25.2.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$4 u^{2}$	-	$5 u$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$	-	$1$

Step 6.2.25.2.5.12

Divide the highest order term in the dividend $- u$ by the highest order term in divisor $u$ .

			-	$4 u^{2}$	-	$5 u$	-	$1$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$	-	$1$

Step 6.2.25.2.5.13

Multiply the new quotient term by the divisor.

			-	$4 u^{2}$	-	$5 u$	-	$1$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$	-	$1$
							-	$u$	-	$1$

Step 6.2.25.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- u - 1$

			-	$4 u^{2}$	-	$5 u$	-	$1$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$	-	$1$
							+	$u$	+	$1$

Step 6.2.25.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$4 u^{2}$	-	$5 u$	-	$1$
$u$	+	$1$	-	$4 u^{3}$	-	$9 u^{2}$	-	$6 u$	-	$1$
			+	$4 u^{3}$	+	$4 u^{2}$
					-	$5 u^{2}$	-	$6 u$
					+	$5 u^{2}$	+	$5 u$
							-	$u$	-	$1$
							+	$u$	+	$1$
										$0$

Step 6.2.25.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$- 4 u^{2} - 5 u - 1$

Step 6.2.25.2.6

Write $- 4 u^{3} - 9 u^{2} - 6 u - 1$ as a set of factors.

$\frac{2 u^{4} ((u + 1) (- 4 u^{2} - 5 u - 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.25.3

Factor by grouping.

$\frac{2 u^{4} (u + 1) (- 4 u - 1) (u + 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.4

Combine exponents.

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Step 6.2.25.4.1

Raise $u + 1$ to the power of $1$ .

$\frac{2 u^{4} ({(u + 1)}^{1} (u + 1)) (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.4.2

Raise $u + 1$ to the power of $1$ .

$\frac{2 u^{4} ({(u + 1)}^{1} {(u + 1)}^{1}) (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.4.3

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

$\frac{2 u^{4} {(u + 1)}^{1 + 1} (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.25.4.4

Add $1$ and $1$ .

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Factor $u$ out of $u^{2} + u$ .

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Step 6.3.1.1

Factor $u$ out of $u^{2}$ .

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u)}^{6}}$

Step 6.3.1.2

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

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u^{1})}^{6}}$

Step 6.3.1.3

Factor $u$ out of $u^{1}$ .

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u \cdot 1)}^{6}}$

Step 6.3.1.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u (u + 1))}^{6}}$

Step 6.3.2

Apply the product rule to $u (u + 1)$ .

$\frac{2 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{u^{6} {(u + 1)}^{6}}$

Step 6.4

Cancel the common factor of $u^{4}$ and $u^{6}$ .

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Step 6.4.1

Factor $u^{4}$ out of $2 u^{4} {(u + 1)}^{2} (- 4 u - 1)$ .

$\frac{u^{4} (2 {(u + 1)}^{2} (- 4 u - 1))}{u^{6} {(u + 1)}^{6}}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $u^{4}$ out of $u^{6} {(u + 1)}^{6}$ .

$\frac{u^{4} (2 {(u + 1)}^{2} (- 4 u - 1))}{u^{4} (u^{2} {(u + 1)}^{6})}$

Step 6.4.2.2

Cancel the common factor.

$\frac{u^{4} (2 {(u + 1)}^{2} (- 4 u - 1))}{u^{4} (u^{2} {(u + 1)}^{6})}$

Step 6.4.2.3

Rewrite the expression.

$\frac{2 {(u + 1)}^{2} (- 4 u - 1)}{u^{2} {(u + 1)}^{6}}$

Step 6.5

Cancel the common factor of ${(u + 1)}^{2}$ and ${(u + 1)}^{6}$ .

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Step 6.5.1

Factor ${(u + 1)}^{2}$ out of $2 {(u + 1)}^{2} (- 4 u - 1)$ .

$\frac{{(u + 1)}^{2} (2 (- 4 u - 1))}{u^{2} {(u + 1)}^{6}}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor ${(u + 1)}^{2}$ out of $u^{2} {(u + 1)}^{6}$ .

$\frac{{(u + 1)}^{2} (2 (- 4 u - 1))}{{(u + 1)}^{2} (u^{2} {(u + 1)}^{4})}$

Step 6.5.2.2

Cancel the common factor.

$\frac{{(u + 1)}^{2} (2 (- 4 u - 1))}{{(u + 1)}^{2} (u^{2} {(u + 1)}^{4})}$

Step 6.5.2.3

Rewrite the expression.

$\frac{2 (- 4 u - 1)}{u^{2} {(u + 1)}^{4}}$

Step 6.6

Factor $- 1$ out of $- 4 u$ .

$\frac{2 (- (4 u) - 1)}{u^{2} {(u + 1)}^{4}}$

Step 6.7

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

$\frac{2 (- (4 u) - 1 (1))}{u^{2} {(u + 1)}^{4}}$

Step 6.8

Factor $- 1$ out of $- (4 u) - 1 (1)$ .

$\frac{2 (- (4 u + 1))}{u^{2} {(u + 1)}^{4}}$

Step 6.9

Rewrite $- (4 u + 1)$ as $- 1 (4 u + 1)$ .

$\frac{2 (- 1 (4 u + 1))}{u^{2} {(u + 1)}^{4}}$

Step 6.10

Move the negative in front of the fraction.

$- \frac{2 (4 u + 1)}{u^{2} {(u + 1)}^{4}}$