Simplify (x^2-3x-10)/(x^2-25)*(2x+10)/(10x+20)

$\frac{x^{2} - 3 x - 10}{x^{2} - 25} \cdot \frac{2 x + 10}{10 x + 20}$

Step 1

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 $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$\frac{(x - 5) (x + 2)}{x^{2} - 25} \cdot \frac{2 x + 10}{10 x + 20}$

Step 2

Simplify the denominator.

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

$\frac{(x - 5) (x + 2)}{x^{2} - 5^{2}} \cdot \frac{2 x + 10}{10 x + 20}$

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 $b = 5$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 x + 10}{10 x + 20}$

Step 3

Simplify terms.

Factor $2$ out of $2 x + 10$ .

Factor $2$ out of $2 x$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 (x) + 10}{10 x + 20}$

Factor $2$ out of $10$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 x + 2 \cdot 5}{10 x + 20}$

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10 x + 20}$

Factor $10$ out of $10 x + 20$ .

Factor $10$ out of $10 x$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10 (x) + 20}$

Factor $10$ out of $20$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10 x + 10 \cdot 2}$

Factor $10$ out of $10 x + 10 \cdot 2$ .

$\frac{(x - 5) (x + 2)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10 (x + 2)}$

Cancel the common factor of $x + 2$ .

Factor $x + 2$ out of $(x - 5) (x + 2)$ .

$\frac{(x + 2) (x - 5)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10 (x + 2)}$

Factor $x + 2$ out of $10 (x + 2)$ .

$\frac{(x + 2) (x - 5)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{(x + 2) \cdot 10}$

Cancel the common factor.

$\frac{(x + 2) (x - 5)}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{(x + 2) \cdot 10}$

Rewrite the expression.

$\frac{x - 5}{(x + 5) (x - 5)} \cdot \frac{2 (x + 5)}{10}$

Cancel the common factor of $x + 5$ .

Factor $x + 5$ out of $2 (x + 5)$ .

$\frac{x - 5}{(x + 5) (x - 5)} \cdot \frac{(x + 5) \cdot 2}{10}$

Cancel the common factor.

$\frac{x - 5}{(x + 5) (x - 5)} \cdot \frac{(x + 5) \cdot 2}{10}$

Rewrite the expression.

$\frac{x - 5}{x - 5} \cdot \frac{2}{10}$

Multiply $\frac{x - 5}{x - 5}$ by $\frac{2}{10}$ .

$\frac{(x - 5) \cdot 2}{(x - 5) \cdot 10}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) \cdot 2}{(x - 5) \cdot 10}$

Rewrite the expression.

$\frac{2}{10}$

Cancel the common factor of $2$ and $10$ .

Factor $2$ out of $2$ .

$\frac{2 (1)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 \cdot 1}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 5}$

Rewrite the expression.

$\frac{1}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{5}$

Decimal Form:

$0.2$

Simplify (x^2-3x-10)/(x^2-25)*(2x+10)/(10x+20)

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