Simplify (6x^5+72x^4+192x^3)/(x^2+13x+40)*(x+2)/(3x^4+12x^3)

$\frac{6 x^{5} + 72 x^{4} + 192 x^{3}}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Step 1

Simplify the numerator.

Factor $6 x^{3}$ out of $6 x^{5} + 72 x^{4} + 192 x^{3}$ .

Factor $6 x^{3}$ out of $6 x^{5}$ .

$\frac{6 x^{3} (x^{2}) + 72 x^{4} + 192 x^{3}}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Factor $6 x^{3}$ out of $72 x^{4}$ .

$\frac{6 x^{3} (x^{2}) + 6 x^{3} (12 x) + 192 x^{3}}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Factor $6 x^{3}$ out of $192 x^{3}$ .

$\frac{6 x^{3} (x^{2}) + 6 x^{3} (12 x) + 6 x^{3} (32)}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Factor $6 x^{3}$ out of $6 x^{3} (x^{2}) + 6 x^{3} (12 x)$ .

$\frac{6 x^{3} (x^{2} + 12 x) + 6 x^{3} (32)}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Factor $6 x^{3}$ out of $6 x^{3} (x^{2} + 12 x) + 6 x^{3} (32)$ .

$\frac{6 x^{3} (x^{2} + 12 x + 32)}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Factor $x^{2} + 12 x + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Write the factored form using these integers.

$\frac{6 x^{3} ((x + 4) (x + 8))}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

$\frac{6 x^{3} (x + 4) (x + 8)}{x^{2} + 13 x + 40} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Step 2

Factor $x^{2} + 13 x + 40$ 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 $40$ and whose sum is $13$ .

$5, 8$

Write the factored form using these integers.

$\frac{6 x^{3} (x + 4) (x + 8)}{(x + 5) (x + 8)} \cdot \frac{x + 2}{3 x^{4} + 12 x^{3}}$

Step 3

Simplify terms.

Factor $3 x^{3}$ out of $3 x^{4} + 12 x^{3}$ .

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

$\frac{6 x^{3} (x + 4) (x + 8)}{(x + 5) (x + 8)} \cdot \frac{x + 2}{3 x^{3} (x) + 12 x^{3}}$

Factor $3 x^{3}$ out of $12 x^{3}$ .

$\frac{6 x^{3} (x + 4) (x + 8)}{(x + 5) (x + 8)} \cdot \frac{x + 2}{3 x^{3} (x) + 3 x^{3} (4)}$

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

$\frac{6 x^{3} (x + 4) (x + 8)}{(x + 5) (x + 8)} \cdot \frac{x + 2}{3 x^{3} (x + 4)}$

Cancel the common factor of $3 x^{3} (x + 4)$ .

Factor $3 x^{3} (x + 4)$ out of $6 x^{3} (x + 4) (x + 8)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(2) (x + 8)}{(x + 5) (x + 8)} \cdot (x + 2)$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{2 (x + 8)}{(x + 5) (x + 8)} \cdot (x + 2)$

Rewrite the expression.

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

Multiply $\frac{2}{x + 5}$ by $x + 2$ .

$\frac{2 (x + 2)}{x + 5}$

Simplify (6x^5+72x^4+192x^3)/(x^2+13x+40)*(x+2)/(3x^4+12x^3)

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