Simplify (x^3+7x^2)/(5x^2+55x+140)*(x^2-16)/((x-4)^2)

$\frac{x^{3} + 7 x^{2}}{5 x^{2} + 55 x + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Step 1

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

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

$\frac{x^{2} x + 7 x^{2}}{5 x^{2} + 55 x + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $x^{2}$ out of $7 x^{2}$ .

$\frac{x^{2} x + x^{2} \cdot 7}{5 x^{2} + 55 x + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 7$ .

$\frac{x^{2} (x + 7)}{5 x^{2} + 55 x + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Step 2

Simplify the denominator.

Factor $5$ out of $5 x^{2} + 55 x + 140$ .

Factor $5$ out of $5 x^{2}$ .

$\frac{x^{2} (x + 7)}{5 (x^{2}) + 55 x + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $5$ out of $55 x$ .

$\frac{x^{2} (x + 7)}{5 (x^{2}) + 5 (11 x) + 140} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $5$ out of $140$ .

$\frac{x^{2} (x + 7)}{5 x^{2} + 5 (11 x) + 5 \cdot 28} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $5$ out of $5 x^{2} + 5 (11 x)$ .

$\frac{x^{2} (x + 7)}{5 (x^{2} + 11 x) + 5 \cdot 28} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $5$ out of $5 (x^{2} + 11 x) + 5 \cdot 28$ .

$\frac{x^{2} (x + 7)}{5 (x^{2} + 11 x + 28)} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Factor $x^{2} + 11 x + 28$ 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 $28$ and whose sum is $11$ .

$4, 7$

Write the factored form using these integers.

$\frac{x^{2} (x + 7)}{5 ((x + 4) (x + 7))} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

$\frac{x^{2} (x + 7)}{5 (x + 4) (x + 7)} \cdot \frac{x^{2} - 16}{{(x - 4)}^{2}}$

Step 3

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 4

Simplify terms.

Combine.

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

Cancel the common factor of $x + 7$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{2} ((x + 4) (x - 4))}{5 (x + 4) {(x - 4)}^{2}}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{x^{2} ((x + 4) (x - 4))}{5 (x + 4) {(x - 4)}^{2}}$

Rewrite the expression.

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

Cancel the common factor of $x - 4$ and ${(x - 4)}^{2}$ .

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

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

Cancel the common factors.

Factor $x - 4$ out of $5 {(x - 4)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x^3+7x^2)/(5x^2+55x+140)*(x^2-16)/((x-4)^2)

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