Simplify ((x^2+5x-24)/(2x+2))÷((3x+24)/(x^2-8x-9))

$\frac{x^{2} + 5 x - 24}{2 x + 2} \div \frac{3 x + 24}{x^{2} - 8 x - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 5 x - 24}{2 x + 2} \cdot \frac{x^{2} - 8 x - 9}{3 x + 24}$

Step 2

Factor $x^{2} + 5 x - 24$ using the AC method.

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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$\frac{(x - 3) (x + 8)}{2 x + 2} \cdot \frac{x^{2} - 8 x - 9}{3 x + 24}$

Step 3

Factor $2$ out of $2 x + 2$ .

Factor $2$ out of $2 x$ .

$\frac{(x - 3) (x + 8)}{2 (x) + 2} \cdot \frac{x^{2} - 8 x - 9}{3 x + 24}$

Factor $2$ out of $2$ .

$\frac{(x - 3) (x + 8)}{2 (x) + 2 (1)} \cdot \frac{x^{2} - 8 x - 9}{3 x + 24}$

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{(x - 3) (x + 8)}{2 (x + 1)} \cdot \frac{x^{2} - 8 x - 9}{3 x + 24}$

Step 4

Factor $x^{2} - 8 x - 9$ using the AC method.

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 $- 9$ and whose sum is $- 8$ .

$- 9, 1$

Write the factored form using these integers.

$\frac{(x - 3) (x + 8)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3 x + 24}$

Step 5

Simplify terms.

Factor $3$ out of $3 x + 24$ .

Factor $3$ out of $3 x$ .

$\frac{(x - 3) (x + 8)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3 (x) + 24}$

Factor $3$ out of $24$ .

$\frac{(x - 3) (x + 8)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3 x + 3 \cdot 8}$

Factor $3$ out of $3 x + 3 \cdot 8$ .

$\frac{(x - 3) (x + 8)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3 (x + 8)}$

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $(x - 3) (x + 8)$ .

$\frac{(x + 8) (x - 3)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3 (x + 8)}$

Factor $x + 8$ out of $3 (x + 8)$ .

$\frac{(x + 8) (x - 3)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{(x + 8) \cdot 3}$

Cancel the common factor.

$\frac{(x + 8) (x - 3)}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{(x + 8) \cdot 3}$

Rewrite the expression.

$\frac{x - 3}{2 (x + 1)} \cdot \frac{(x - 9) (x + 1)}{3}$

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $2 (x + 1)$ .

$\frac{x - 3}{(x + 1) \cdot 2} \cdot \frac{(x - 9) (x + 1)}{3}$

Factor $x + 1$ out of $(x - 9) (x + 1)$ .

$\frac{x - 3}{(x + 1) \cdot 2} \cdot \frac{(x + 1) (x - 9)}{3}$

Cancel the common factor.

$\frac{x - 3}{(x + 1) \cdot 2} \cdot \frac{(x + 1) (x - 9)}{3}$

Rewrite the expression.

$\frac{x - 3}{2} \cdot \frac{x - 9}{3}$

Multiply $\frac{x - 3}{2}$ by $\frac{x - 9}{3}$ .

$\frac{(x - 3) (x - 9)}{2 \cdot 3}$

Multiply $2$ by $3$ .

$\frac{(x - 3) (x - 9)}{6}$

Simplify ((x^2+5x-24)/(2x+2))÷((3x+24)/(x^2-8x-9))

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