Simplify (x^4+x^3)/(x^2+3x+2)*(15x^2-60)/(15x^5-75x^4+90x^3)

$\frac{x^{4} + x^{3}}{x^{2} + 3 x + 2} \cdot \frac{15 x^{2} - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

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

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

$\frac{x^{3} x + x^{3}}{x^{2} + 3 x + 2} \cdot \frac{15 x^{2} - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Multiply by $1$ .

$\frac{x^{3} x + x^{3} \cdot 1}{x^{2} + 3 x + 2} \cdot \frac{15 x^{2} - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot 1$ .

$\frac{x^{3} (x + 1)}{x^{2} + 3 x + 2} \cdot \frac{15 x^{2} - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

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

$1, 2$

Write the factored form using these integers.

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 x^{2} - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Simplify the numerator.

Factor $15$ out of $15 x^{2} - 60$ .

Factor $15$ out of $15 x^{2}$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x^{2}) - 60}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Factor $15$ out of $- 60$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 x^{2} + 15 \cdot - 4}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Factor $15$ out of $15 x^{2} + 15 \cdot - 4$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x^{2} - 4)}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Rewrite $4$ as $2^{2}$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x^{2} - 2^{2})}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

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 $b = 2$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{5} - 75 x^{4} + 90 x^{3}}$

Simplify the denominator.

Factor $15 x^{3}$ out of $15 x^{5} - 75 x^{4} + 90 x^{3}$ .

Factor $15 x^{3}$ out of $15 x^{5}$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{3} (x^{2}) - 75 x^{4} + 90 x^{3}}$

Factor $15 x^{3}$ out of $- 75 x^{4}$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{3} (x^{2}) + 15 x^{3} (- 5 x) + 90 x^{3}}$

Factor $15 x^{3}$ out of $90 x^{3}$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{3} (x^{2}) + 15 x^{3} (- 5 x) + 15 x^{3} (6)}$

Factor $15 x^{3}$ out of $15 x^{3} (x^{2}) + 15 x^{3} (- 5 x)$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{3} (x^{2} - 5 x) + 15 x^{3} (6)}$

Factor $15 x^{3}$ out of $15 x^{3} (x^{2} - 5 x) + 15 x^{3} (6)$ .

$\frac{x^{3} (x + 1)}{(x + 1) (x + 2)} \cdot \frac{15 (x + 2) (x - 2)}{15 x^{3} (x^{2} - 5 x + 6)}$

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

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

Simplify terms.

Combine.

$\frac{x^{3} (x + 1) (15 (x + 2) (x - 2))}{(x + 1) (x + 2) (15 x^{3} (x - 3) (x - 2))}$

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

Cancel the common factor.

$\frac{x^{3} (x + 1) (15 (x + 2) (x - 2))}{(x + 1) (x + 2) (15 x^{3} (x - 3) (x - 2))}$

Rewrite the expression.

$\frac{(x + 1) (15 (x + 2) (x - 2))}{(x + 1) (x + 2) ((15 (x - 3)) (x - 2))}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (15 (x + 2) (x - 2))}{(x + 1) (x + 2) ((15 (x - 3)) (x - 2))}$

Rewrite the expression.

$\frac{15 (x + 2) (x - 2)}{(x + 2) ((15 (x - 3)) (x - 2))}$

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{15 (x + 2) (x - 2)}{(x + 2) (15 (x - 3) (x - 2))}$

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 3}$

Simplify (x^4+x^3)/(x^2+3x+2)*(15x^2-60)/(15x^5-75x^4+90x^3)

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