Algebra Examples

$\frac{\frac{1}{a^{2} + 7 a + 12} + \frac{1}{a^{2} + a - 6}}{\frac{1}{a^{2} + 2 a - 8} + \frac{1}{a^{2} + 5 a + 4}}$

Step 1

Multiply the numerator and denominator of the fraction by $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

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Step 1.1

Multiply $\frac{\frac{1}{a^{2} + 7 a + 12} + \frac{1}{a^{2} + a - 6}}{\frac{1}{a^{2} + 2 a - 8} + \frac{1}{a^{2} + 5 a + 4}}$ by $\frac{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)}$ .

$\frac{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)} \cdot \frac{\frac{1}{a^{2} + 7 a + 12} + \frac{1}{a^{2} + a - 6}}{\frac{1}{a^{2} + 2 a - 8} + \frac{1}{a^{2} + 5 a + 4}}$

Step 1.2

Combine.

$\frac{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) (\frac{1}{a^{2} + 7 a + 12} + \frac{1}{a^{2} + a - 6})}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) (\frac{1}{a^{2} + 2 a - 8} + \frac{1}{a^{2} + 5 a + 4})}$

Step 2

Apply the distributive property.

$\frac{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 7 a + 12} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + a - 6}}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $a^{2} + 7 a + 12$ .

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Step 3.1.1

Factor $a^{2} + 7 a + 12$ out of $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

$\frac{(a^{2} + 7 a + 12) (((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)) \frac{1}{a^{2} + 7 a + 12} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + a - 6}}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.1.2

Cancel the common factor.

Step 3.1.3

Rewrite the expression.

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + a - 6}}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.2

Cancel the common factor of $a^{2} + a - 6$ .

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Step 3.2.1

Factor $a^{2} + a - 6$ out of $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + (a^{2} + a - 6) (((a^{2} + 7 a + 12) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)) \frac{1}{a^{2} + a - 6}}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.2.2

Cancel the common factor.

Step 3.2.3

Rewrite the expression.

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + ((a^{2} + 7 a + 12) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.3

Cancel the common factor of $a^{2} + 2 a - 8$ .

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Step 3.3.1

Factor $a^{2} + 2 a - 8$ out of $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + ((a^{2} + 7 a + 12) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)}{(a^{2} + 2 a - 8) ((a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4)) \frac{1}{a^{2} + 2 a - 8} + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.3.2

Cancel the common factor.

Step 3.3.3

Rewrite the expression.

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + ((a^{2} + 7 a + 12) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.4

Cancel the common factor of $a^{2} + 5 a + 4$ .

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Step 3.4.1

Factor $a^{2} + 5 a + 4$ out of $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

$\frac{((a^{2} + a - 6) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4) + ((a^{2} + 7 a + 12) (a^{2} + 2 a - 8)) (a^{2} + 5 a + 4)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 5 a + 4) ((a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)) \frac{1}{a^{2} + 5 a + 4}}$

Step 3.4.2

Cancel the common factor.

Step 3.4.3

Rewrite the expression.

Step 4

Simplify the numerator.

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Step 4.1

Factor $(a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ out of $(a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

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Step 4.1.1

Factor $(a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ out of $(a^{2} + a - 6) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

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

Step 4.1.2

Factor $(a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ out of $(a^{2} + 7 a + 12) (a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ .

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

Step 4.1.3

Factor $(a^{2} + 2 a - 8) (a^{2} + 5 a + 4)$ out of $(a^{2} + 2 a - 8) (a^{2} + 5 a + 4) (a^{2} + a - 6) + (a^{2} + 2 a - 8) (a^{2} + 5 a + 4) (a^{2} + 7 a + 12)$ .

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

Step 4.2

Factor $a^{2} + 2 a - 8$ using the AC method.

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Step 4.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 4.2.2

Write the factored form using these integers.

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

Step 4.3

Factor $a^{2} + 5 a + 4$ using the AC method.

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Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $5$ .

$1, 4$

Step 4.3.2

Write the factored form using these integers.

$\frac{(a - 2) (a + 4) ((a + 1) (a + 4)) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

$\frac{(a - 2) (a + 4) (a + 1) (a + 4) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.4

Combine exponents.

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Step 4.4.1

Raise $a + 4$ to the power of $1$ .

$\frac{(a - 2) ({(a + 4)}^{1} (a + 4)) (a + 1) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.4.2

Raise $a + 4$ to the power of $1$ .

$\frac{(a - 2) ({(a + 4)}^{1} {(a + 4)}^{1}) (a + 1) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.4.3

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

$\frac{(a - 2) {(a + 4)}^{1 + 1} (a + 1) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.4.4

Add $1$ and $1$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (a^{2} + a - 6 + a^{2} + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.5

Add $a^{2}$ and $a^{2}$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 a^{2} + a - 6 + 7 a + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.6

Add $a$ and $7 a$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 a^{2} + 8 a - 6 + 12)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.7

Add $- 6$ and $12$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 a^{2} + 8 a + 6)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.8

Factor $2$ out of $2 a^{2} + 8 a + 6$ .

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Step 4.8.1

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

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 (a^{2}) + 8 a + 6)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.8.2

Factor $2$ out of $8 a$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 (a^{2}) + 2 (4 a) + 6)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.8.3

Factor $2$ out of $6$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 a^{2} + 2 (4 a) + 2 \cdot 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.8.4

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

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 (a^{2} + 4 a) + 2 \cdot 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.8.5

Factor $2$ out of $2 (a^{2} + 4 a) + 2 \cdot 3$ .

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) (2 (a^{2} + 4 a + 3))}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.9

Factor.

$\frac{(a - 2) {(a + 4)}^{2} (a + 1) \cdot 2 (a + 1) (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.10

Combine exponents.

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Step 4.10.1

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

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 ({(a + 1)}^{1} (a + 1)) (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.10.2

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

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 ({(a + 1)}^{1} {(a + 1)}^{1}) (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.10.3

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

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 {(a + 1)}^{1 + 1} (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 4.10.4

Add $1$ and $1$ .

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 {(a + 1)}^{2} (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $(a^{2} + 7 a + 12) (a^{2} + a - 6)$ out of $(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4) + (a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 2 a - 8)$ .

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 {(a + 1)}^{2} (a + 3)}{(a^{2} + 7 a + 12) (a^{2} + a - 6) (a^{2} + 5 a + 4 + a^{2} + 2 a - 8)}$

Step 5.2

Factor $a^{2} + 7 a + 12$ using the AC method.

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Step 5.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $7$ .

$3, 4$

Step 5.2.2

Write the factored form using these integers.

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 {(a + 1)}^{2} (a + 3)}{(a + 3) (a + 4) (a^{2} + a - 6) (a^{2} + 5 a + 4 + a^{2} + 2 a - 8)}$

Step 5.3

Factor $a^{2} + a - 6$ using the AC method.

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Step 5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 5.3.2

Write the factored form using these integers.

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

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

Step 5.4

Combine exponents.

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Step 5.4.1

Raise $a + 3$ to the power of $1$ .

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

Step 5.4.2

Raise $a + 3$ to the power of $1$ .

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

Step 5.4.3

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

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 5.5

Add $a^{2}$ and $a^{2}$ .

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

Step 5.6

Add $5 a$ and $2 a$ .

$\frac{(a - 2) {(a + 4)}^{2} \cdot 2 {(a + 1)}^{2} (a + 3)}{{(a + 3)}^{2} (a + 4) (a - 2) (2 a^{2} + 7 a + 4 - 8)}$

Step 5.7

Subtract $8$ from $4$ .

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

Step 5.8

Factor by grouping.

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Step 5.8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = 7$ .

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Step 5.8.1.1

Factor $7$ out of $7 a$ .

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

Step 5.8.1.2

Rewrite $7$ as $- 1$ plus $8$

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

Step 5.8.1.3

Apply the distributive property.

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

Step 5.8.2

Factor out the greatest common factor from each group.

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Step 5.8.2.1

Group the first two terms and the last two terms.

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

Step 5.8.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.8.3

Factor the polynomial by factoring out the greatest common factor, $2 a - 1$ .

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

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

Step 5.9

Combine exponents.

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Step 5.9.1

Raise $a + 4$ to the power of $1$ .

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

Step 5.9.2

Raise $a + 4$ to the power of $1$ .

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

Step 5.9.3

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

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

Step 5.9.4

Add $1$ and $1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $a - 2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of ${(a + 4)}^{2}$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

Cancel the common factor of $a + 3$ and ${(a + 3)}^{2}$ .

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Step 6.3.1

Factor $a + 3$ out of $2 {(a + 1)}^{2} (a + 3)$ .

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

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $a + 3$ out of ${(a + 3)}^{2} (2 a - 1)$ .

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

Step 6.3.2.2

Cancel the common factor.

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

Step 6.3.2.3

Rewrite the expression.

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