Simplify (x^2+14x+40)/(9x^4+135x^3+450x^2)*(6x^2+84x+270)/(x^2+13x+36)

$\frac{x^{2} + 14 x + 40}{9 x^{4} + 135 x^{3} + 450 x^{2}} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Step 1

Factor $x^{2} + 14 x + 40$ 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 $14$ .

$4, 10$

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{9 x^{4} + 135 x^{3} + 450 x^{2}} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Step 2

Simplify the denominator.

Factor $9 x^{2}$ out of $9 x^{4} + 135 x^{3} + 450 x^{2}$ .

Factor $9 x^{2}$ out of $9 x^{4}$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x^{2}) + 135 x^{3} + 450 x^{2}} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $9 x^{2}$ out of $135 x^{3}$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x^{2}) + 9 x^{2} (15 x) + 450 x^{2}} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $9 x^{2}$ out of $450 x^{2}$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x^{2}) + 9 x^{2} (15 x) + 9 x^{2} (50)} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $9 x^{2}$ out of $9 x^{2} (x^{2}) + 9 x^{2} (15 x)$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x^{2} + 15 x) + 9 x^{2} (50)} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $9 x^{2}$ out of $9 x^{2} (x^{2} + 15 x) + 9 x^{2} (50)$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x^{2} + 15 x + 50)} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $x^{2} + 15 x + 50$ 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 $50$ and whose sum is $15$ .

$5, 10$

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{9 x^{2} ((x + 5) (x + 10))} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 x^{2} + 84 x + 270}{x^{2} + 13 x + 36}$

Step 3

Simplify the numerator.

Factor $6$ out of $6 x^{2} + 84 x + 270$ .

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

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x^{2}) + 84 x + 270}{x^{2} + 13 x + 36}$

Factor $6$ out of $84 x$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x^{2}) + 6 (14 x) + 270}{x^{2} + 13 x + 36}$

Factor $6$ out of $270$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 x^{2} + 6 (14 x) + 6 \cdot 45}{x^{2} + 13 x + 36}$

Factor $6$ out of $6 x^{2} + 6 (14 x)$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x^{2} + 14 x) + 6 \cdot 45}{x^{2} + 13 x + 36}$

Factor $6$ out of $6 (x^{2} + 14 x) + 6 \cdot 45$ .

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x^{2} + 14 x + 45)}{x^{2} + 13 x + 36}$

Factor $x^{2} + 14 x + 45$ 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 $45$ and whose sum is $14$ .

$5, 9$

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 ((x + 5) (x + 9))}{x^{2} + 13 x + 36}$

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x + 5) (x + 9)}{x^{2} + 13 x + 36}$

Step 4

Factor $x^{2} + 13 x + 36$ 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 $36$ and whose sum is $13$ .

$4, 9$

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{9 x^{2} (x + 5) (x + 10)} \cdot \frac{6 (x + 5) (x + 9)}{(x + 4) (x + 9)}$

Step 5

Simplify terms.

Combine.

$\frac{(x + 4) (x + 10) (6 (x + 5) (x + 9))}{9 x^{2} (x + 5) (x + 10) ((x + 4) (x + 9))}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x + 4) (x + 10) (6 (x + 5) (x + 9))}{9 x^{2} (x + 5) (x + 10) ((x + 4) (x + 9))}$

Rewrite the expression.

$\frac{(x + 10) (6 (x + 5) (x + 9))}{9 x^{2} (x + 5) (x + 10) (x + 9)}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{(x + 10) (6 (x + 5) (x + 9))}{9 x^{2} (x + 5) (x + 10) (x + 9)}$

Rewrite the expression.

$\frac{6 (x + 5) (x + 9)}{9 x^{2} (x + 5) (x + 9)}$

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6 (x + 5) (x + 9)$ .

$\frac{3 (2 (x + 5) (x + 9))}{9 x^{2} (x + 5) (x + 9)}$

Cancel the common factors.

Factor $3$ out of $9 x^{2} (x + 5) (x + 9)$ .

$\frac{3 (2 (x + 5) (x + 9))}{3 (3 x^{2} (x + 5) (x + 9))}$

Cancel the common factor.

$\frac{3 (2 (x + 5) (x + 9))}{3 (3 x^{2} (x + 5) (x + 9))}$

Rewrite the expression.

$\frac{2 (x + 5) (x + 9)}{3 x^{2} (x + 5) (x + 9)}$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{2 (x + 5) (x + 9)}{3 x^{2} (x + 5) (x + 9)}$

Rewrite the expression.

$\frac{2 (x + 9)}{3 x^{2} (x + 9)}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{2 (x + 9)}{3 x^{2} (x + 9)}$

Rewrite the expression.

$\frac{2}{3 x^{2}}$

Simplify (x^2+14x+40)/(9x^4+135x^3+450x^2)*(6x^2+84x+270)/(x^2+13x+36)

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