Simplify (6x^4-384x^2)/(x+3)*(x^2-9)/(6x^2+30x-144)

$\frac{6 x^{4} - 384 x^{2}}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

Step 1

Simplify the numerator.

Factor $6 x^{2}$ out of $6 x^{4} - 384 x^{2}$ .

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

$\frac{6 x^{2} (x^{2}) - 384 x^{2}}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

Factor $6 x^{2}$ out of $- 384 x^{2}$ .

$\frac{6 x^{2} (x^{2}) + 6 x^{2} (- 64)}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2}) + 6 x^{2} (- 64)$ .

$\frac{6 x^{2} (x^{2} - 64)}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

Rewrite $64$ as $8^{2}$ .

$\frac{6 x^{2} (x^{2} - 8^{2})}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

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 $b = 8$ .

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{x^{2} - 9}{6 x^{2} + 30 x - 144}$

Step 2

Simplify the numerator.

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

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{x^{2} - 3^{2}}{6 x^{2} + 30 x - 144}$

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 $b = 3$ .

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 x^{2} + 30 x - 144}$

Step 3

Simplify the denominator.

Factor $6$ out of $6 x^{2} + 30 x - 144$ .

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

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 (x^{2}) + 30 x - 144}$

Factor $6$ out of $30 x$ .

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 (x^{2}) + 6 (5 x) - 144}$

Factor $6$ out of $- 144$ .

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 x^{2} + 6 (5 x) + 6 \cdot - 24}$

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

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 (x^{2} + 5 x) + 6 \cdot - 24}$

Factor $6$ out of $6 (x^{2} + 5 x) + 6 \cdot - 24$ .

$\frac{6 x^{2} (x + 8) (x - 8)}{x + 3} \cdot \frac{(x + 3) (x - 3)}{6 (x^{2} + 5 x - 24)}$

Factor $x^{2} + 5 x - 24$ 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 $5$ .

$- 3, 8$

Write the factored form using these integers.

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

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

Step 4

Simplify terms.

Cancel the common factor of $6 (x + 8)$ .

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

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

Factor $6 (x + 8)$ out of $6 (x - 3) (x + 8)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{x - 3}{x - 3}$ and $x^{2}$ .

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

$x^{2} (x - 8)$

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 8$

Step 5

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} \cdot - 8$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot - 8$

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 8$

Step 6

Move $- 8$ to the left of $x^{2}$ .

$x^{3} - 8 x^{2}$

Simplify (6x^4-384x^2)/(x+3)*(x^2-9)/(6x^2+30x-144)

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