Simplify (x^2-7x+12)/(x^2-8x+15)2/(x^2-49)(3x^2-36x+105)/(x-4)

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 x^{2} - 36 x + 105}{x - 4}$

Simplify the numerator.

Factor $3$ out of $3 x^{2} - 36 x + 105$ .

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

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x^{2}) - 36 x + 105}{x - 4}$

Factor $3$ out of $- 36 x$ .

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x^{2}) + 3 (- 12 x) + 105}{x - 4}$

Factor $3$ out of $105$ .

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 x^{2} + 3 (- 12 x) + 3 \cdot 35}{x - 4}$

Factor $3$ out of $3 x^{2} + 3 (- 12 x)$ .

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x^{2} - 12 x) + 3 \cdot 35}{x - 4}$

Factor $3$ out of $3 (x^{2} - 12 x) + 3 \cdot 35$ .

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x^{2} - 12 x + 35)}{x - 4}$

Factor $x^{2} - 12 x + 35$ 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 $- 12$ .

$- 7, - 5$

Write the factored form using these integers.

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 ((x - 7) (x - 5))}{x - 4}$

$\frac{x^{2} - 7 x + 12}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

$\frac{(x - 4) (x - 3)}{x^{2} - 8 x + 15} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$\frac{(x - 4) (x - 3)}{(x - 5) (x - 3)} \cdot \frac{2}{x^{2} - 49} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Simplify the denominator.

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

$\frac{(x - 4) (x - 3)}{(x - 5) (x - 3)} \cdot \frac{2}{x^{2} - 7^{2}} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

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 $b = 7$ .

$\frac{(x - 4) (x - 3)}{(x - 5) (x - 3)} \cdot \frac{2}{(x + 7) (x - 7)} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Simplify terms.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(x - 4) (x - 3)}{(x - 5) (x - 3)} \cdot \frac{2}{(x + 7) (x - 7)} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Rewrite the expression.

$\frac{x - 4}{x - 5} \cdot \frac{2}{(x + 7) (x - 7)} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Multiply $\frac{x - 4}{x - 5}$ by $\frac{2}{(x + 7) (x - 7)}$ .

$\frac{(x - 4) \cdot 2}{(x - 5) ((x + 7) (x - 7))} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Cancel the common factor of $x - 4$ .

Factor $x - 4$ out of $(x - 4) \cdot 2$ .

$\frac{(x - 4) (2)}{(x - 5) (x + 7) (x - 7)} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Cancel the common factor.

$\frac{(x - 4) \cdot 2}{(x - 5) (x + 7) (x - 7)} \cdot \frac{3 (x - 7) (x - 5)}{x - 4}$

Rewrite the expression.

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

Cancel the common factor of $(x - 7) (x - 5)$ .

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

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

Factor $(x - 7) (x - 5)$ out of $3 (x - 7) (x - 5)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2}{x + 7} \cdot 3$

Combine $\frac{2}{x + 7}$ and $3$ .

$\frac{2 \cdot 3}{x + 7}$

Multiply $2$ by $3$ .

$\frac{6}{x + 7}$

Simplify (x^2-7x+12)/(x^2-8x+15)*2/(x^2-49)*(3x^2-36x+105)/(x-4)

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Simplify (x^2-7x+12)/(x^2-8x+15)2/(x^2-49)(3x^2-36x+105)/(x-4)