Simplify (4x^3+24x^2-64x)/(x^2+3x-10)*(x^2-25)/(6x^4+18x^3-240x^2)

$\frac{4 x^{3} + 24 x^{2} - 64 x}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Step 1

Simplify the numerator.

Factor $4 x$ out of $4 x^{3} + 24 x^{2} - 64 x$ .

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

$\frac{4 x (x^{2}) + 24 x^{2} - 64 x}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Factor $4 x$ out of $24 x^{2}$ .

$\frac{4 x (x^{2}) + 4 x (6 x) - 64 x}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Factor $4 x$ out of $- 64 x$ .

$\frac{4 x (x^{2}) + 4 x (6 x) + 4 x (- 16)}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

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

$\frac{4 x (x^{2} + 6 x) + 4 x (- 16)}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Factor $4 x$ out of $4 x (x^{2} + 6 x) + 4 x (- 16)$ .

$\frac{4 x (x^{2} + 6 x - 16)}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Factor $x^{2} + 6 x - 16$ 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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Write the factored form using these integers.

$\frac{4 x ((x - 2) (x + 8))}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

$\frac{4 x (x - 2) (x + 8)}{x^{2} + 3 x - 10} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Step 2

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{x^{2} - 25}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Step 3

Simplify the numerator.

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

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{x^{2} - 5^{2}}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

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 $b = 5$ .

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{4} + 18 x^{3} - 240 x^{2}}$

Step 4

Simplify the denominator.

Factor $6 x^{2}$ out of $6 x^{4} + 18 x^{3} - 240 x^{2}$ .

Factor $6 x^{2}$ out of $6 x^{4}$ .

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x^{2}) + 18 x^{3} - 240 x^{2}}$

Factor $6 x^{2}$ out of $18 x^{3}$ .

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x^{2}) + 6 x^{2} (3 x) - 240 x^{2}}$

Factor $6 x^{2}$ out of $- 240 x^{2}$ .

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x^{2}) + 6 x^{2} (3 x) + 6 x^{2} (- 40)}$

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

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x^{2} + 3 x) + 6 x^{2} (- 40)}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2} + 3 x) + 6 x^{2} (- 40)$ .

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x^{2} + 3 x - 40)}$

Factor $x^{2} + 3 x - 40$ 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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Write the factored form using these integers.

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} ((x - 5) (x + 8))}$

$\frac{4 x (x - 2) (x + 8)}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{6 x^{2} (x - 5) (x + 8)}$

Step 5

Simplify terms.

Combine.

$\frac{4 x (x - 2) (x + 8) ((x + 5) (x - 5))}{(x - 2) (x + 5) (6 x^{2} (x - 5) (x + 8))}$

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4 x (x - 2) (x + 8) ((x + 5) (x - 5))$ .

$\frac{2 (2 x (x - 2) (x + 8) ((x + 5) (x - 5)))}{(x - 2) (x + 5) (6 x^{2} (x - 5) (x + 8))}$

Cancel the common factors.

Factor $2$ out of $(x - 2) (x + 5) (6 x^{2} (x - 5) (x + 8))$ .

$\frac{2 (2 x (x - 2) (x + 8) ((x + 5) (x - 5)))}{2 ((x - 2) (x + 5) (3 x^{2} (x - 5) (x + 8)))}$

Cancel the common factor.

$\frac{2 (2 x (x - 2) (x + 8) ((x + 5) (x - 5)))}{2 ((x - 2) (x + 5) (3 x^{2} (x - 5) (x + 8)))}$

Rewrite the expression.

$\frac{2 x (x - 2) (x + 8) ((x + 5) (x - 5))}{(x - 2) (x + 5) (3 x^{2} (x - 5) (x + 8))}$

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

Factor $x$ out of $2 x (x - 2) (x + 8) ((x + 5) (x - 5))$ .

$\frac{x (((2 (x - 2)) (x + 8)) ((x + 5) (x - 5)))}{(x - 2) (x + 5) (3 x^{2} (x - 5) (x + 8))}$

Cancel the common factors.

Factor $x$ out of $(x - 2) (x + 5) (3 x^{2} (x - 5) (x + 8))$ .

$\frac{x (((2 (x - 2)) (x + 8)) ((x + 5) (x - 5)))}{x ((x - 2) (x + 5) (3 x (x - 5) (x + 8)))}$

Cancel the common factor.

$\frac{x (((2 (x - 2)) (x + 8)) ((x + 5) (x - 5)))}{x ((x - 2) (x + 5) (3 x (x - 5) (x + 8)))}$

Rewrite the expression.

$\frac{((2 (x - 2)) (x + 8)) ((x + 5) (x - 5))}{(x - 2) (x + 5) (3 x (x - 5) (x + 8))}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{2 (x - 2) (x + 8) ((x + 5) (x - 5))}{(x - 2) (x + 5) (3 x (x - 5) (x + 8))}$

Rewrite the expression.

$\frac{((2) (x + 8)) ((x + 5) (x - 5))}{(x + 5) (3 x (x - 5) (x + 8))}$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{2 (x + 8) ((x + 5) (x - 5))}{(x + 5) (3 x (x - 5) (x + 8))}$

Rewrite the expression.

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

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2}{3 x}$

Simplify (4x^3+24x^2-64x)/(x^2+3x-10)*(x^2-25)/(6x^4+18x^3-240x^2)

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