Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = {(y - 1)}^{4}$ and $g (y) = {(y^{2} + 2 y)}^{5}$ .

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

Step 2

Multiply the exponents in ${({(y^{2} + 2 y)}^{5})}^{2}$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Multiply $5$ by $2$ .

$\frac{{(y^{2} + 2 y)}^{5} \frac{d}{d y} [{(y - 1)}^{4}] - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{4}$ and $g (y) = y - 1$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u_{1}$ as $y - 1$ .

$\frac{{(y^{2} + 2 y)}^{5} (\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d y} [y - 1]) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 3.2

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

$\frac{{(y^{2} + 2 y)}^{5} (4 {u_{1}}^{3} \frac{d}{d y} [y - 1]) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 3.3

Replace all occurrences of $u_{1}$ with $y - 1$ .

$\frac{{(y^{2} + 2 y)}^{5} (4 {(y - 1)}^{3} \frac{d}{d y} [y - 1]) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

Move $4$ to the left of ${(y^{2} + 2 y)}^{5}$ .

$\frac{4 \cdot {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} \frac{d}{d y} [y - 1] - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4.2

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} (\frac{d}{d y} [y] + \frac{d}{d y} [- 1]) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4.3

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} (1 + \frac{d}{d y} [- 1]) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4.4

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} (1 + 0) - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4.5

Simplify the expression.

Tap for more steps...

Step 4.5.1

Add $1$ and $0$ .

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} \cdot 1 - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 4.5.2

Multiply $4$ by $1$ .

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - {(y - 1)}^{4} \frac{d}{d y} [{(y^{2} + 2 y)}^{5}]}{{(y^{2} + 2 y)}^{10}}$

Step 5

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

Tap for more steps...

Step 5.1

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - {(y - 1)}^{4} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{10}}$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - {(y - 1)}^{4} (5 {u_{2}}^{4} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{10}}$

Step 5.3

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - {(y - 1)}^{4} (5 {(y^{2} + 2 y)}^{4} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{10}}$

Step 6

Differentiate.

Tap for more steps...

Step 6.1

Multiply $5$ by $- 1$ .

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} \frac{d}{d y} [y^{2} + 2 y])}{{(y^{2} + 2 y)}^{10}}$

Step 6.2

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (\frac{d}{d y} [y^{2}] + \frac{d}{d y} [2 y]))}{{(y^{2} + 2 y)}^{10}}$

Step 6.3

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (2 y + \frac{d}{d y} [2 y]))}{{(y^{2} + 2 y)}^{10}}$

Step 6.4

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (2 y + 2 \frac{d}{d y} [y]))}{{(y^{2} + 2 y)}^{10}}$

Step 6.5

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

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (2 y + 2 \cdot 1))}{{(y^{2} + 2 y)}^{10}}$

Step 6.6

Multiply $2$ by $1$ .

$\frac{4 {(y^{2} + 2 y)}^{5} {(y - 1)}^{3} - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

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

Tap for more steps...

Step 7.1.1.1

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

$\frac{{(y^{2} + 2 y)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y)) - 5 {(y - 1)}^{4} ({(y^{2} + 2 y)}^{4} (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.1.2

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

$\frac{{(y^{2} + 2 y)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y)) + {(y^{2} + 2 y)}^{4} {(y - 1)}^{3} (- 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.1.3

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

$\frac{{(y^{2} + 2 y)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.2

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

Tap for more steps...

Step 7.1.2.1

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

$\frac{{(y \cdot y + 2 y)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.2.2

Factor $y$ out of $2 y$ .

$\frac{{(y \cdot y + y \cdot 2)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.2.3

Factor $y$ out of $y \cdot y + y \cdot 2$ .

$\frac{{(y (y + 2))}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.3

Apply the product rule to $y (y + 2)$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 (y^{2} + 2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.4

Apply the distributive property.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 4 (2 y) - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.5

Multiply $2$ by $4$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 (y - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.6

Apply the distributive property.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y + (- 5 y - 5 \cdot - 1) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.7

Multiply $- 5$ by $- 1$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y + (- 5 y + 5) (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.8

Expand $(- 5 y + 5) (2 y + 2)$ using the FOIL Method.

Tap for more steps...

Step 7.1.8.1

Apply the distributive property.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 y (2 y + 2) + 5 (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.8.2

Apply the distributive property.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 y (2 y) - 5 y \cdot 2 + 5 (2 y + 2))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.8.3

Apply the distributive property.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 y (2 y) - 5 y \cdot 2 + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9

Simplify and combine like terms.

Tap for more steps...

Step 7.1.9.1

Simplify each term.

Tap for more steps...

Step 7.1.9.1.1

Rewrite using the commutative property of multiplication.

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 \cdot 2 y \cdot y - 5 y \cdot 2 + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 7.1.9.1.2.1

Move $y$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 \cdot 2 (y \cdot y) - 5 y \cdot 2 + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.2.2

Multiply $y$ by $y$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 5 \cdot 2 y^{2} - 5 y \cdot 2 + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.3

Multiply $- 5$ by $2$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 10 y^{2} - 5 y \cdot 2 + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.4

Multiply $2$ by $- 5$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 10 y^{2} - 10 y + 5 (2 y) + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.5

Multiply $2$ by $5$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 10 y^{2} - 10 y + 10 y + 5 \cdot 2)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.1.6

Multiply $5$ by $2$ .

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

Step 7.1.9.2

Add $- 10 y$ and $10 y$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (4 y^{2} + 8 y - 10 y^{2} + 0 + 10)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.9.3

Add $- 10 y^{2}$ and $0$ .

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

Step 7.1.10

Subtract $10 y^{2}$ from $4 y^{2}$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 6 y^{2} + 8 y + 10)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.11

Factor $2$ out of $- 6 y^{2} + 8 y + 10$ .

Tap for more steps...

Step 7.1.11.1

Factor $2$ out of $- 6 y^{2}$ .

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

Step 7.1.11.2

Factor $2$ out of $8 y$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (2 (- 3 y^{2}) + 2 (4 y) + 10)}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.11.3

Factor $2$ out of $10$ .

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (2 (- 3 y^{2}) + 2 (4 y) + 2 (5))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.11.4

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

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (2 (- 3 y^{2} + 4 y) + 2 (5))}{{(y^{2} + 2 y)}^{10}}$

Step 7.1.11.5

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

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (2 (- 3 y^{2} + 4 y + 5))}{{(y^{2} + 2 y)}^{10}}$

$\frac{y^{4} {(y + 2)}^{4} {(y - 1)}^{3} \cdot 2 (- 3 y^{2} + 4 y + 5)}{{(y^{2} + 2 y)}^{10}}$

Step 7.2

Move $2$ to the left of $y^{4} {(y + 2)}^{4} {(y - 1)}^{3}$ .

$\frac{2 y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{{(y^{2} + 2 y)}^{10}}$

Step 7.3

Simplify the denominator.

Tap for more steps...

Step 7.3.1

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

Tap for more steps...

Step 7.3.1.1

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

$\frac{2 y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{{(y \cdot y + 2 y)}^{10}}$

Step 7.3.1.2

Factor $y$ out of $2 y$ .

$\frac{2 y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{{(y \cdot y + y \cdot 2)}^{10}}$

Step 7.3.1.3

Factor $y$ out of $y \cdot y + y \cdot 2$ .

$\frac{2 y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{{(y (y + 2))}^{10}}$

Step 7.3.2

Apply the product rule to $y (y + 2)$ .

$\frac{2 y^{4} {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{y^{10} {(y + 2)}^{10}}$

Step 7.4

Cancel the common factor of $y^{4}$ and $y^{10}$ .

Tap for more steps...

Step 7.4.1

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

$\frac{y^{4} (2 {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5))}{y^{10} {(y + 2)}^{10}}$

Step 7.4.2

Cancel the common factors.

Tap for more steps...

Step 7.4.2.1

Factor $y^{4}$ out of $y^{10} {(y + 2)}^{10}$ .

$\frac{y^{4} (2 {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5))}{y^{4} (y^{6} {(y + 2)}^{10})}$

Step 7.4.2.2

Cancel the common factor.

$\frac{y^{4} (2 {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5))}{y^{4} (y^{6} {(y + 2)}^{10})}$

Step 7.4.2.3

Rewrite the expression.

$\frac{2 {(y + 2)}^{4} {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5)}{y^{6} {(y + 2)}^{10}}$

Step 7.5

Cancel the common factor of ${(y + 2)}^{4}$ and ${(y + 2)}^{10}$ .

Tap for more steps...

Step 7.5.1

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

$\frac{{(y + 2)}^{4} (2 {(y - 1)}^{3} (- 3 y^{2} + 4 y + 5))}{y^{6} {(y + 2)}^{10}}$

Step 7.5.2

Cancel the common factors.

Tap for more steps...

Step 7.5.2.1

Factor ${(y + 2)}^{4}$ out of $y^{6} {(y + 2)}^{10}$ .

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

Step 7.5.2.2

Cancel the common factor.

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

Step 7.5.2.3

Rewrite the expression.

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

Step 7.6

Factor $- 1$ out of $- 3 y^{2}$ .

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

Step 7.7

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

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

Step 7.8

Factor $- 1$ out of $- (3 y^{2}) - (- 4 y)$ .

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

Step 7.9

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 7.10

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

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

Step 7.11

Rewrite $- (3 y^{2} - 4 y - 5)$ as $- 1 (3 y^{2} - 4 y - 5)$ .

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

Step 7.12

Move the negative in front of the fraction.

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

Step 7.13

Reorder factors in $- \frac{(2 {(y - 1)}^{3}) (3 y^{2} - 4 y - 5)}{y^{6} {(y + 2)}^{6}}$ .

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