Simplify (2x^2-50)/(x-5)*(x+4)/(x^4+9x^3+20x^2)

$\frac{2 x^{2} - 50}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

Step 1

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 50$ .

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

$\frac{2 (x^{2}) - 50}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

Factor $2$ out of $- 50$ .

$\frac{2 x^{2} + 2 \cdot - 25}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 25$ .

$\frac{2 (x^{2} - 25)}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

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

$\frac{2 (x^{2} - 5^{2})}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

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 $b = 5$ .

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{4} + 9 x^{3} + 20 x^{2}}$

Step 2

Simplify the denominator.

Factor $x^{2}$ out of $x^{4} + 9 x^{3} + 20 x^{2}$ .

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

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{2} x^{2} + 9 x^{3} + 20 x^{2}}$

Factor $x^{2}$ out of $9 x^{3}$ .

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{2} x^{2} + x^{2} (9 x) + 20 x^{2}}$

Factor $x^{2}$ out of $20 x^{2}$ .

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{2} x^{2} + x^{2} (9 x) + x^{2} \cdot 20}$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (9 x)$ .

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{2} (x^{2} + 9 x) + x^{2} \cdot 20}$

Factor $x^{2}$ out of $x^{2} (x^{2} + 9 x) + x^{2} \cdot 20$ .

$\frac{2 (x + 5) (x - 5)}{x - 5} \cdot \frac{x + 4}{x^{2} (x^{2} + 9 x + 20)}$

Factor $x^{2} + 9 x + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Write the factored form using these integers.

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

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

Step 3

Simplify terms.

Cancel the common factor of $x + 5$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 (x + 4)}{x^{2} (x + 4)}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{2 (x + 4)}{x^{2} (x + 4)}$

Rewrite the expression.

$\frac{2}{x^{2}}$

Simplify (2x^2-50)/(x-5)*(x+4)/(x^4+9x^3+20x^2)

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