Simplify (6x^3-294x)/(3x^4+39x^3+126x^2)*(x^2-36)/(x-7)

$\frac{6 x^{3} - 294 x}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Simplify the numerator.

Factor $6 x$ out of $6 x^{3} - 294 x$ .

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

$\frac{6 x (x^{2}) - 294 x}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $6 x$ out of $- 294 x$ .

$\frac{6 x (x^{2}) + 6 x (- 49)}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 49)$ .

$\frac{6 x (x^{2} - 49)}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Rewrite $49$ as $7^{2}$ .

$\frac{6 x (x^{2} - 7^{2})}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

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 $b = 7$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{4} + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Simplify the denominator.

Factor $3 x^{2}$ out of $3 x^{4} + 39 x^{3} + 126 x^{2}$ .

Factor $3 x^{2}$ out of $3 x^{4}$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x^{2}) + 39 x^{3} + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $3 x^{2}$ out of $39 x^{3}$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x^{2}) + 3 x^{2} (13 x) + 126 x^{2}} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $3 x^{2}$ out of $126 x^{2}$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x^{2}) + 3 x^{2} (13 x) + 3 x^{2} (42)} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $3 x^{2}$ out of $3 x^{2} (x^{2}) + 3 x^{2} (13 x)$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x^{2} + 13 x) + 3 x^{2} (42)} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $3 x^{2}$ out of $3 x^{2} (x^{2} + 13 x) + 3 x^{2} (42)$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x^{2} + 13 x + 42)} \cdot \frac{x^{2} - 36}{x - 7}$

Factor $x^{2} + 13 x + 42$ 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 $42$ and whose sum is $13$ .

$6, 7$

Write the factored form using these integers.

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} ((x + 6) (x + 7))} \cdot \frac{x^{2} - 36}{x - 7}$

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x + 6) (x + 7)} \cdot \frac{x^{2} - 36}{x - 7}$

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x + 6) (x + 7)} \cdot \frac{x^{2} - 6^{2}}{x - 7}$

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 $b = 6$ .

$\frac{6 x (x + 7) (x - 7)}{3 x^{2} (x + 6) (x + 7)} \cdot \frac{(x + 6) (x - 6)}{x - 7}$

Simplify terms.

Cancel the common factor of $x - 7$ .

Factor $x - 7$ out of $6 x (x + 7) (x - 7)$ .

$\frac{(x - 7) (6 x (x + 7))}{3 x^{2} (x + 6) (x + 7)} \cdot \frac{(x + 6) (x - 6)}{x - 7}$

Cancel the common factor.

$\frac{(x - 7) (6 x (x + 7))}{3 x^{2} (x + 6) (x + 7)} \cdot \frac{(x + 6) (x - 6)}{x - 7}$

Rewrite the expression.

$\frac{6 x (x + 7)}{3 x^{2} (x + 6) (x + 7)} \cdot ((x + 6) (x - 6))$

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $3 x^{2} (x + 6) (x + 7)$ .

$\frac{6 x (x + 7)}{(x + 6) (3 x^{2} (x + 7))} \cdot ((x + 6) (x - 6))$

Cancel the common factor.

$\frac{6 x (x + 7)}{(x + 6) (3 x^{2} (x + 7))} \cdot ((x + 6) (x - 6))$

Rewrite the expression.

$\frac{6 x (x + 7)}{3 x^{2} (x + 7)} \cdot (x - 6)$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 x (x + 7)$ .

$\frac{3 (2 x (x + 7))}{3 x^{2} (x + 7)} \cdot (x - 6)$

Cancel the common factors.

Factor $3$ out of $3 x^{2} (x + 7)$ .

$\frac{3 (2 x (x + 7))}{3 (x^{2} (x + 7))} \cdot (x - 6)$

Cancel the common factor.

$\frac{3 (2 x (x + 7))}{3 (x^{2} (x + 7))} \cdot (x - 6)$

Rewrite the expression.

$\frac{2 x (x + 7)}{x^{2} (x + 7)} \cdot (x - 6)$

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

Factor $x$ out of $2 x (x + 7)$ .

$\frac{x (2 (x + 7))}{x^{2} (x + 7)} \cdot (x - 6)$

Cancel the common factors.

Factor $x$ out of $x^{2} (x + 7)$ .

$\frac{x (2 (x + 7))}{x (x (x + 7))} \cdot (x - 6)$

Cancel the common factor.

$\frac{x (2 (x + 7))}{x (x (x + 7))} \cdot (x - 6)$

Rewrite the expression.

$\frac{2 (x + 7)}{x (x + 7)} \cdot (x - 6)$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{2 (x + 7)}{x (x + 7)} \cdot (x - 6)$

Rewrite the expression.

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

Multiply $\frac{2}{x}$ by $x - 6$ .

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

Simplify (6x^3-294x)/(3x^4+39x^3+126x^2)*(x^2-36)/(x-7)

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