Simplify (2x^5-50x^3)/(x^2+4x-45)*(x^2-81)/(6x^2-24x-270)

$\frac{2 x^{5} - 50 x^{3}}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Simplify the numerator.

Factor $2 x^{3}$ out of $2 x^{5} - 50 x^{3}$ .

Factor $2 x^{3}$ out of $2 x^{5}$ .

$\frac{2 x^{3} (x^{2}) - 50 x^{3}}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Factor $2 x^{3}$ out of $- 50 x^{3}$ .

$\frac{2 x^{3} (x^{2}) + 2 x^{3} (- 25)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Factor $2 x^{3}$ out of $2 x^{3} (x^{2}) + 2 x^{3} (- 25)$ .

$\frac{2 x^{3} (x^{2} - 25)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

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

$\frac{2 x^{3} (x^{2} - 5^{2})}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

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 $b = 5$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{x^{2} + 4 x - 45} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Factor $x^{2} + 4 x - 45$ 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 $- 45$ and whose sum is $4$ .

$- 5, 9$

Write the factored form using these integers.

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{x^{2} - 81}{6 x^{2} - 24 x - 270}$

Simplify the numerator.

Rewrite $81$ as $9^{2}$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{x^{2} - 9^{2}}{6 x^{2} - 24 x - 270}$

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 $b = 9$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 x^{2} - 24 x - 270}$

Simplify the denominator.

Factor $6$ out of $6 x^{2} - 24 x - 270$ .

Factor $6$ out of $6 x^{2}$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2}) - 24 x - 270}$

Factor $6$ out of $- 24 x$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2}) + 6 (- 4 x) - 270}$

Factor $6$ out of $- 270$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 x^{2} + 6 (- 4 x) + 6 \cdot - 45}$

Factor $6$ out of $6 x^{2} + 6 (- 4 x)$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2} - 4 x) + 6 \cdot - 45}$

Factor $6$ out of $6 (x^{2} - 4 x) + 6 \cdot - 45$ .

$\frac{2 x^{3} (x + 5) (x - 5)}{(x - 5) (x + 9)} \cdot \frac{(x + 9) (x - 9)}{6 (x^{2} - 4 x - 45)}$

Factor $x^{2} - 4 x - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Write the factored form using these integers.

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

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

Simplify terms.

Combine.

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

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 x^{3} (x + 5) (x - 5) ((x + 9) (x - 9))$ .

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

Cancel the common factors.

Factor $2$ out of $(x - 5) (x + 9) (6 (x - 9) (x + 5))$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{((x^{3} (x + 5)) (x - 5)) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9) (x + 5))}$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{x^{3} (x + 5) (x - 5) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9) (x + 5))}$

Rewrite the expression.

$\frac{((x^{3}) (x - 5)) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9))}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x^{3} (x - 5) ((x + 9) (x - 9))}{(x - 5) (x + 9) (3 (x - 9))}$

Rewrite the expression.

$\frac{(x^{3}) ((x + 9) (x - 9))}{(x + 9) (3 (x - 9))}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{x^{3} ((x + 9) (x - 9))}{(x + 9) (3 (x - 9))}$

Rewrite the expression.

$\frac{(x^{3}) (x - 9)}{3 (x - 9)}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{x^{3} (x - 9)}{3 (x - 9)}$

Rewrite the expression.

$\frac{x^{3}}{3}$

Simplify (2x^5-50x^3)/(x^2+4x-45)*(x^2-81)/(6x^2-24x-270)

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