Simplify ((6x^3+12x^2-210x)/(x^2-49)*(6x)/(2x^3-50x))÷((x+7)/(x^2-49))

$\frac{6 x^{3} + 12 x^{2} - 210 x}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \div \frac{x + 7}{x^{2} - 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 x^{3} + 12 x^{2} - 210 x}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Step 2

Simplify the numerator.

Factor $6 x$ out of $6 x^{3} + 12 x^{2} - 210 x$ .

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

$\frac{6 x (x^{2}) + 12 x^{2} - 210 x}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Factor $6 x$ out of $12 x^{2}$ .

$\frac{6 x (x^{2}) + 6 x (2 x) - 210 x}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Factor $6 x$ out of $- 210 x$ .

$\frac{6 x (x^{2}) + 6 x (2 x) + 6 x (- 35)}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

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

$\frac{6 x (x^{2} + 2 x) + 6 x (- 35)}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Factor $6 x$ out of $6 x (x^{2} + 2 x) + 6 x (- 35)$ .

$\frac{6 x (x^{2} + 2 x - 35)}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Write the factored form using these integers.

$\frac{6 x ((x - 5) (x + 7))}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

$\frac{6 x (x - 5) (x + 7)}{x^{2} - 49} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Step 3

Simplify the denominator.

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

$\frac{6 x (x - 5) (x + 7)}{x^{2} - 7^{2}} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

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 $b = 7$ .

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x^{3} - 50 x} \frac{x^{2} - 49}{x + 7}$

Step 4

Simplify the denominator.

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

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

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x^{2}) - 50 x} \frac{x^{2} - 49}{x + 7}$

Factor $2 x$ out of $- 50 x$ .

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x^{2}) + 2 x (- 25)} \frac{x^{2} - 49}{x + 7}$

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

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x^{2} - 25)} \frac{x^{2} - 49}{x + 7}$

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

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x^{2} - 5^{2})} \frac{x^{2} - 49}{x + 7}$

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 $b = 5$ .

$\frac{6 x (x - 5) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x + 5) (x - 5)} \frac{x^{2} - 49}{x + 7}$

Step 5

Simplify terms.

Cancel the common factor of $2 x (x - 5)$ .

Factor $2 x (x - 5)$ out of $6 x (x - 5) (x + 7)$ .

$\frac{2 x (x - 5) ((3) (x + 7))}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x + 5) (x - 5)} \frac{x^{2} - 49}{x + 7}$

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

$\frac{2 x (x - 5) ((3) (x + 7))}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x - 5) (x + 5)} \frac{x^{2} - 49}{x + 7}$

Cancel the common factor.

$\frac{2 x (x - 5) ((3) (x + 7))}{(x + 7) (x - 7)} \cdot \frac{6 x}{2 x (x - 5) (x + 5)} \frac{x^{2} - 49}{x + 7}$

Rewrite the expression.

$\frac{(3) (x + 7)}{(x + 7) (x - 7)} \cdot \frac{6 x}{x + 5} \frac{x^{2} - 49}{x + 7}$

Multiply $\frac{(3) (x + 7)}{(x + 7) (x - 7)}$ by $\frac{6 x}{x + 5}$ .

$\frac{(3) (x + 7) (6 x)}{(x + 7) (x - 7) (x + 5)} \cdot \frac{x^{2} - 49}{x + 7}$

Multiply $6$ by $3$ .

$\frac{18 (x + 7) x}{(x + 7) (x - 7) (x + 5)} \cdot \frac{x^{2} - 49}{x + 7}$

Cancel the common factor of $x + 7$ .

Factor $x + 7$ out of $18 (x + 7) x$ .

$\frac{(x + 7) (18 x)}{(x + 7) (x - 7) (x + 5)} \cdot \frac{x^{2} - 49}{x + 7}$

Cancel the common factor.

$\frac{(x + 7) (18 x)}{(x + 7) (x - 7) (x + 5)} \cdot \frac{x^{2} - 49}{x + 7}$

Rewrite the expression.

$\frac{18 x}{(x + 7) (x - 7) (x + 5)} (x^{2} - 49)$

Multiply $\frac{18 x}{(x + 7) (x - 7) (x + 5)}$ by $x^{2} - 49$ .

$\frac{18 x (x^{2} - 49)}{(x + 7) (x - 7) (x + 5)}$

Step 6

Simplify the numerator.

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

$\frac{18 x (x^{2} - 7^{2})}{(x + 7) (x - 7) (x + 5)}$

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 $b = 7$ .

$\frac{18 x (x + 7) (x - 7)}{(x + 7) (x - 7) (x + 5)}$

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{18 x (x + 7) (x - 7)}{(x + 7) (x - 7) (x + 5)}$

Rewrite the expression.

$\frac{18 x (x - 7)}{(x - 7) (x + 5)}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{18 x (x - 7)}{(x - 7) (x + 5)}$

Rewrite the expression.

$\frac{18 x}{x + 5}$

Simplify ((6x^3+12x^2-210x)/(x^2-49)*(6x)/(2x^3-50x))÷((x+7)/(x^2-49))

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