Simplify (x^5-7x^4-30x^3)/(x^2+10x+21)*(4x^2-196)/(4x^3-68x^2+280x)

$\frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Simplify the numerator.

Factor $x^{3}$ out of $x^{5} - 7 x^{4} - 30 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$\frac{x^{3} x^{2} - 7 x^{4} - 30 x^{3}}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $x^{3}$ out of $- 7 x^{4}$ .

$\frac{x^{3} x^{2} + x^{3} (- 7 x) - 30 x^{3}}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $x^{3}$ out of $- 30 x^{3}$ .

$\frac{x^{3} x^{2} + x^{3} (- 7 x) + x^{3} \cdot - 30}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

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

$\frac{x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $x^{3}$ out of $x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30$ .

$\frac{x^{3} (x^{2} - 7 x - 30)}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $x^{2} - 7 x - 30$ 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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Write the factored form using these integers.

$\frac{x^{3} ((x - 10) (x + 3))}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

$\frac{x^{3} (x - 10) (x + 3)}{x^{2} + 10 x + 21} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $x^{2} + 10 x + 21$ 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 $21$ and whose sum is $10$ .

$3, 7$

Write the factored form using these integers.

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 x^{2} - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Simplify the numerator.

Factor $4$ out of $4 x^{2} - 196$ .

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

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x^{2}) - 196}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $4$ out of $- 196$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 x^{2} + 4 \cdot - 49}{4 x^{3} - 68 x^{2} + 280 x}$

Factor $4$ out of $4 x^{2} + 4 \cdot - 49$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x^{2} - 49)}{4 x^{3} - 68 x^{2} + 280 x}$

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

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x^{2} - 7^{2})}{4 x^{3} - 68 x^{2} + 280 x}$

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 $b = 7$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x^{3} - 68 x^{2} + 280 x}$

Simplify the denominator.

Factor $4 x$ out of $4 x^{3} - 68 x^{2} + 280 x$ .

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

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x^{2}) - 68 x^{2} + 280 x}$

Factor $4 x$ out of $- 68 x^{2}$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x^{2}) + 4 x (- 17 x) + 280 x}$

Factor $4 x$ out of $280 x$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x^{2}) + 4 x (- 17 x) + 4 x (70)}$

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 17 x)$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x^{2} - 17 x) + 4 x (70)}$

Factor $4 x$ out of $4 x (x^{2} - 17 x) + 4 x (70)$ .

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x^{2} - 17 x + 70)}$

Factor $x^{2} - 17 x + 70$ 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 $70$ and whose sum is $- 17$ .

$- 10, - 7$

Write the factored form using these integers.

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x ((x - 10) (x - 7))}$

$\frac{x^{3} (x - 10) (x + 3)}{(x + 3) (x + 7)} \cdot \frac{4 (x + 7) (x - 7)}{4 x (x - 10) (x - 7)}$

Simplify terms.

Combine.

$\frac{x^{3} (x - 10) (x + 3) (4 (x + 7) (x - 7))}{(x + 3) (x + 7) (4 x (x - 10) (x - 7))}$

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $x^{3} (x - 10) (x + 3) (4 (x + 7) (x - 7))$ .

$\frac{x (((x^{2} (x - 10)) (x + 3)) (4 (x + 7) (x - 7)))}{(x + 3) (x + 7) (4 x (x - 10) (x - 7))}$

Cancel the common factors.

Factor $x$ out of $(x + 3) (x + 7) (4 x (x - 10) (x - 7))$ .

$\frac{x (((x^{2} (x - 10)) (x + 3)) (4 (x + 7) (x - 7)))}{x ((x + 3) (x + 7) ((4 (x - 10)) (x - 7)))}$

Cancel the common factor.

$\frac{x (((x^{2} (x - 10)) (x + 3)) (4 (x + 7) (x - 7)))}{x ((x + 3) (x + 7) ((4 (x - 10)) (x - 7)))}$

Rewrite the expression.

$\frac{((x^{2} (x - 10)) (x + 3)) (4 (x + 7) (x - 7))}{(x + 3) (x + 7) ((4 (x - 10)) (x - 7))}$

Cancel the common factor of $x - 10$ .

Cancel the common factor.

$\frac{x^{2} (x - 10) (x + 3) (4 (x + 7) (x - 7))}{(x + 3) (x + 7) (4 (x - 10) (x - 7))}$

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x^{2}) (4 (x + 7) (x - 7))}{(x + 7) ((4) (x - 7))}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{x^{2} (4 (x + 7) (x - 7))}{(x + 7) (4 (x - 7))}$

Rewrite the expression.

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

Cancel the common factor of $x + 7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 7$ .

Cancel the common factor.

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

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

$x^{2}$

Simplify (x^5-7x^4-30x^3)/(x^2+10x+21)*(4x^2-196)/(4x^3-68x^2+280x)

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