Simplify ((3x-15)/(x^2-25))÷((12(x+1))/(x^2+3x-10))

$\frac{3 x - 15}{x^{2} - 25} \div \frac{12 (x + 1)}{x^{2} + 3 x - 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x - 15}{x^{2} - 25} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Step 2

Factor $3$ out of $3 x - 15$ .

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 15}{x^{2} - 25} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Factor $3$ out of $- 15$ .

$\frac{3 x + 3 \cdot - 5}{x^{2} - 25} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Factor $3$ out of $3 x + 3 \cdot - 5$ .

$\frac{3 (x - 5)}{x^{2} - 25} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Step 3

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$\frac{3 (x - 5)}{x^{2} - 5^{2}} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{3 (x - 5)}{(x + 5) (x - 5)} \cdot \frac{x^{2} + 3 x - 10}{12 (x + 1)}$

Step 4

Simplify terms.

Cancel the common factor of $3$ .

Factor $3$ out of $12 (x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x - 5}{(x + 5) (x - 5)}$ by $\frac{x^{2} + 3 x - 10}{4 (x + 1)}$ .

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{(x - 2) (x + 5)}{(x + 5) \cdot 4 (x + 1)}$

Step 6

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{(x - 2) (x + 5)}{(x + 5) \cdot 4 (x + 1)}$

Rewrite the expression.

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

Simplify ((3x-15)/(x^2-25))÷((12(x+1))/(x^2+3x-10))

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