Simplify (x^3-9x)/(x+3)*(5x^2-50x+120)/(10x^2-90x+180)

$\frac{x^{3} - 9 x}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

Simplify the numerator.

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

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

$\frac{x \cdot x^{2} - 9 x}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

Factor $x$ out of $- 9 x$ .

$\frac{x \cdot x^{2} + x \cdot - 9}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$\frac{x (x^{2} - 9)}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

Rewrite $9$ as $3^{2}$ .

$\frac{x (x^{2} - 3^{2})}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

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 $b = 3$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 x^{2} - 50 x + 120}{10 x^{2} - 90 x + 180}$

Simplify the numerator.

Factor $5$ out of $5 x^{2} - 50 x + 120$ .

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

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x^{2}) - 50 x + 120}{10 x^{2} - 90 x + 180}$

Factor $5$ out of $- 50 x$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x^{2}) + 5 (- 10 x) + 120}{10 x^{2} - 90 x + 180}$

Factor $5$ out of $120$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 x^{2} + 5 (- 10 x) + 5 \cdot 24}{10 x^{2} - 90 x + 180}$

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

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x^{2} - 10 x) + 5 \cdot 24}{10 x^{2} - 90 x + 180}$

Factor $5$ out of $5 (x^{2} - 10 x) + 5 \cdot 24$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x^{2} - 10 x + 24)}{10 x^{2} - 90 x + 180}$

Factor $x^{2} - 10 x + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 ((x - 6) (x - 4))}{10 x^{2} - 90 x + 180}$

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 x^{2} - 90 x + 180}$

Simplify the denominator.

Factor $10$ out of $10 x^{2} - 90 x + 180$ .

Factor $10$ out of $10 x^{2}$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x^{2}) - 90 x + 180}$

Factor $10$ out of $- 90 x$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x^{2}) + 10 (- 9 x) + 180}$

Factor $10$ out of $180$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 x^{2} + 10 (- 9 x) + 10 \cdot 18}$

Factor $10$ out of $10 x^{2} + 10 (- 9 x)$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x^{2} - 9 x) + 10 \cdot 18}$

Factor $10$ out of $10 (x^{2} - 9 x) + 10 \cdot 18$ .

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x^{2} - 9 x + 18)}$

Factor $x^{2} - 9 x + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 ((x - 6) (x - 3))}$

$\frac{x (x + 3) (x - 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x - 6) (x - 3)}$

Simplify terms.

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $x (x + 3) (x - 3)$ .

$\frac{(x - 3) (x (x + 3))}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x - 6) (x - 3)}$

Factor $x - 3$ out of $10 (x - 6) (x - 3)$ .

$\frac{(x - 3) (x (x + 3))}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{(x - 3) (10 (x - 6))}$

Cancel the common factor.

$\frac{(x - 3) (x (x + 3))}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{(x - 3) (10 (x - 6))}$

Rewrite the expression.

$\frac{x (x + 3)}{x + 3} \cdot \frac{5 (x - 6) (x - 4)}{10 (x - 6)}$

Multiply $\frac{x (x + 3)}{x + 3}$ by $\frac{5 (x - 6) (x - 4)}{10 (x - 6)}$ .

$\frac{x (x + 3) (5 (x - 6) (x - 4))}{(x + 3) (10 (x - 6))}$

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{x (x + 3) (5 (x - 6) (x - 4))}{(x + 3) (10 (x - 6))}$

Rewrite the expression.

$\frac{x (5 (x - 6) (x - 4))}{10 (x - 6)}$

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $x (5 (x - 6) (x - 4))$ .

$\frac{5 (x ((x - 6) (x - 4)))}{10 (x - 6)}$

Cancel the common factors.

Factor $5$ out of $10 (x - 6)$ .

$\frac{5 (x ((x - 6) (x - 4)))}{5 (2 (x - 6))}$

Cancel the common factor.

$\frac{5 (x ((x - 6) (x - 4)))}{5 (2 (x - 6))}$

Rewrite the expression.

$\frac{x ((x - 6) (x - 4))}{2 (x - 6)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{x ((x - 6) (x - 4))}{2 (x - 6)}$

Rewrite the expression.

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

Simplify (x^3-9x)/(x+3)*(5x^2-50x+120)/(10x^2-90x+180)

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