Finite Math Examples

$\frac{4 s^{2} + 20 s + 24}{16 s^{2} - 32 s - 128}$

Step 1

Cancel the common factor of $4 s^{2} + 20 s + 24$ and $16 s^{2} - 32 s - 128$ .

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Step 1.1

Factor $4$ out of $4 s^{2}$ .

$\frac{4 (s^{2}) + 20 s + 24}{16 s^{2} - 32 s - 128}$

Step 1.2

Factor $4$ out of $20 s$ .

$\frac{4 (s^{2}) + 4 (5 s) + 24}{16 s^{2} - 32 s - 128}$

Step 1.3

Factor $4$ out of $4 (s^{2}) + 4 (5 s)$ .

$\frac{4 (s^{2} + 5 s) + 24}{16 s^{2} - 32 s - 128}$

Step 1.4

Factor $4$ out of $24$ .

$\frac{4 (s^{2} + 5 s) + 4 \cdot 6}{16 s^{2} - 32 s - 128}$

Step 1.5

Factor $4$ out of $4 (s^{2} + 5 s) + 4 \cdot 6$ .

$\frac{4 (s^{2} + 5 s + 6)}{16 s^{2} - 32 s - 128}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $4$ out of $16 s^{2}$ .

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2}) - 32 s - 128}$

Step 1.6.2

Factor $4$ out of $- 32 s$ .

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2}) + 4 (- 8 s) - 128}$

Step 1.6.3

Factor $4$ out of $4 (4 s^{2}) + 4 (- 8 s)$ .

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2} - 8 s) - 128}$

Step 1.6.4

Factor $4$ out of $- 128$ .

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2} - 8 s) + 4 (- 32)}$

Step 1.6.5

Factor $4$ out of $4 (4 s^{2} - 8 s) + 4 (- 32)$ .

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2} - 8 s - 32)}$

Step 1.6.6

Cancel the common factor.

$\frac{4 (s^{2} + 5 s + 6)}{4 (4 s^{2} - 8 s - 32)}$

Step 1.6.7

Rewrite the expression.

$\frac{s^{2} + 5 s + 6}{4 s^{2} - 8 s - 32}$

Step 2

Factor $s^{2} + 5 s + 6$ using the AC method.

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Step 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 $6$ and whose sum is $5$ .

$2, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(s + 2) (s + 3)}{4 s^{2} - 8 s - 32}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $4$ out of $4 s^{2} - 8 s - 32$ .

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Step 3.1.1

Factor $4$ out of $4 s^{2}$ .

$\frac{(s + 2) (s + 3)}{4 (s^{2}) - 8 s - 32}$

Step 3.1.2

Factor $4$ out of $- 8 s$ .

$\frac{(s + 2) (s + 3)}{4 (s^{2}) + 4 (- 2 s) - 32}$

Step 3.1.3

Factor $4$ out of $- 32$ .

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

Step 3.1.4

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

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

Step 3.1.5

Factor $4$ out of $4 (s^{2} - 2 s) + 4 \cdot - 8$ .

$\frac{(s + 2) (s + 3)}{4 (s^{2} - 2 s - 8)}$

Step 3.2

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

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Step 3.2.1

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 $- 2$ .

$- 4, 2$

Step 3.2.2

Write the factored form using these integers.

$\frac{(s + 2) (s + 3)}{4 ((s - 4) (s + 2))}$

$\frac{(s + 2) (s + 3)}{4 (s - 4) (s + 2)}$

Step 4

Cancel the common factor of $s + 2$ .

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Step 4.1

Cancel the common factor.

$\frac{(s + 2) (s + 3)}{4 (s - 4) (s + 2)}$

Step 4.2

Rewrite the expression.

$\frac{s + 3}{4 (s - 4)}$