Simplify (5x^4-500x^2)/(15x+75)*(x^2+11x+30)/(x^5+16x^4+60x^3)

$\frac{5 x^{4} - 500 x^{2}}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Simplify the numerator.

Factor $5 x^{2}$ out of $5 x^{4} - 500 x^{2}$ .

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

$\frac{5 x^{2} (x^{2}) - 500 x^{2}}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $5 x^{2}$ out of $- 500 x^{2}$ .

$\frac{5 x^{2} (x^{2}) + 5 x^{2} (- 100)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $5 x^{2}$ out of $5 x^{2} (x^{2}) + 5 x^{2} (- 100)$ .

$\frac{5 x^{2} (x^{2} - 100)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Rewrite $100$ as $10^{2}$ .

$\frac{5 x^{2} (x^{2} - 10^{2})}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

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 $b = 10$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $15$ out of $15 x + 75$ .

Factor $15$ out of $15 x$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x) + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $15$ out of $75$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 x + 15 \cdot 5} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $15$ out of $15 x + 15 \cdot 5$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Factor $x^{2} + 11 x + 30$ 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 $30$ and whose sum is $11$ .

$5, 6$

Write the factored form using these integers.

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{5} + 16 x^{4} + 60 x^{3}}$

Simplify the denominator.

Factor $x^{3}$ out of $x^{5} + 16 x^{4} + 60 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + 16 x^{4} + 60 x^{3}}$

Factor $x^{3}$ out of $16 x^{4}$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + x^{3} (16 x) + 60 x^{3}}$

Factor $x^{3}$ out of $60 x^{3}$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + x^{3} (16 x) + x^{3} \cdot 60}$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (16 x)$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x^{2} + 16 x) + x^{3} \cdot 60}$

Factor $x^{3}$ out of $x^{3} (x^{2} + 16 x) + x^{3} \cdot 60$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x^{2} + 16 x + 60)}$

Factor $x^{2} + 16 x + 60$ 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 $60$ and whose sum is $16$ .

$6, 10$

Write the factored form using these integers.

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} ((x + 6) (x + 10))}$

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x + 6) (x + 10)}$

Simplify terms.

Combine.

$\frac{5 x^{2} (x + 10) (x - 10) ((x + 5) (x + 6))}{15 (x + 5) (x^{3} (x + 6) (x + 10))}$

Cancel the common factor of $5$ and $15$ .

Factor $5$ out of $5 x^{2} (x + 10) (x - 10) ((x + 5) (x + 6))$ .

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{15 (x + 5) (x^{3} (x + 6) (x + 10))}$

Cancel the common factors.

Factor $5$ out of $15 (x + 5) (x^{3} (x + 6) (x + 10))$ .

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{5 (3 (x + 5) (x^{3} (x + 6) (x + 10)))}$

Cancel the common factor.

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{5 (3 (x + 5) (x^{3} (x + 6) (x + 10)))}$

Rewrite the expression.

$\frac{((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6))}{3 (x + 5) (x^{3} (x + 6) (x + 10))}$

Cancel the common factor of $x^{2}$ and $x^{3}$ .

Factor $x^{2}$ out of $((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6))$ .

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{3 (x + 5) (x^{3} (x + 6) (x + 10))}$

Cancel the common factors.

Factor $x^{2}$ out of $3 (x + 5) (x^{3} (x + 6) (x + 10))$ .

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{x^{2} (3 (x + 5) ((x (x + 6)) (x + 10)))}$

Cancel the common factor.

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{x^{2} (3 (x + 5) ((x (x + 6)) (x + 10)))}$

Rewrite the expression.

$\frac{((x + 10) (x - 10)) ((x + 5) (x + 6))}{3 (x + 5) ((x (x + 6)) (x + 10))}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{(x + 10) (x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6) (x + 10))}$

Rewrite the expression.

$\frac{(x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6))}$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{(x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6))}$

Rewrite the expression.

$\frac{(x - 10) (x + 6)}{3 (x (x + 6))}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x - 10) (x + 6)}{3 (x (x + 6))}$

Rewrite the expression.

$\frac{x - 10}{3 (x)}$

$\frac{x - 10}{3 x}$

Simplify (5x^4-500x^2)/(15x+75)*(x^2+11x+30)/(x^5+16x^4+60x^3)

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