Simplify (x^4+14x^3+40x^2)/(x^2+9x+20)*(3x+15)/(x^2+11x+10)

$\frac{x^{4} + 14 x^{3} + 40 x^{2}}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Step 1

Simplify the numerator.

Factor $x^{2}$ out of $x^{4} + 14 x^{3} + 40 x^{2}$ .

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

$\frac{x^{2} x^{2} + 14 x^{3} + 40 x^{2}}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Factor $x^{2}$ out of $14 x^{3}$ .

$\frac{x^{2} x^{2} + x^{2} (14 x) + 40 x^{2}}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Factor $x^{2}$ out of $40 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} (14 x) + x^{2} \cdot 40}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

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

$\frac{x^{2} (x^{2} + 14 x) + x^{2} \cdot 40}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Factor $x^{2}$ out of $x^{2} (x^{2} + 14 x) + x^{2} \cdot 40$ .

$\frac{x^{2} (x^{2} + 14 x + 40)}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

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.

$\frac{x^{2} ((x + 4) (x + 10))}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

$\frac{x^{2} (x + 4) (x + 10)}{x^{2} + 9 x + 20} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Step 2

Factor $x^{2} + 9 x + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Write the factored form using these integers.

$\frac{x^{2} (x + 4) (x + 10)}{(x + 4) (x + 5)} \cdot \frac{3 x + 15}{x^{2} + 11 x + 10}$

Step 3

Factor $3$ out of $3 x + 15$ .

Factor $3$ out of $3 x$ .

$\frac{x^{2} (x + 4) (x + 10)}{(x + 4) (x + 5)} \cdot \frac{3 (x) + 15}{x^{2} + 11 x + 10}$

Factor $3$ out of $15$ .

$\frac{x^{2} (x + 4) (x + 10)}{(x + 4) (x + 5)} \cdot \frac{3 x + 3 \cdot 5}{x^{2} + 11 x + 10}$

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{x^{2} (x + 4) (x + 10)}{(x + 4) (x + 5)} \cdot \frac{3 (x + 5)}{x^{2} + 11 x + 10}$

Step 4

Factor $x^{2} + 11 x + 10$ 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 $10$ and whose sum is $11$ .

$1, 10$

Write the factored form using these integers.

$\frac{x^{2} (x + 4) (x + 10)}{(x + 4) (x + 5)} \cdot \frac{3 (x + 5)}{(x + 1) (x + 10)}$

Step 5

Simplify terms.

Combine.

$\frac{x^{2} (x + 4) (x + 10) (3 (x + 5))}{(x + 4) (x + 5) ((x + 1) (x + 10))}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{x^{2} (x + 4) (x + 10) (3 (x + 5))}{(x + 4) (x + 5) ((x + 1) (x + 10))}$

Rewrite the expression.

$\frac{(x^{2} (x + 10)) (3 (x + 5))}{(x + 5) ((x + 1) (x + 10))}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{x^{2} (x + 10) (3 (x + 5))}{(x + 5) ((x + 1) (x + 10))}$

Rewrite the expression.

$\frac{(x^{2}) (3 (x + 5))}{(x + 5) (x + 1)}$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{x^{2} (3 (x + 5))}{(x + 5) (x + 1)}$

Rewrite the expression.

$\frac{(x^{2}) \cdot (3)}{x + 1}$

Move $3$ to the left of $x^{2}$ .

$\frac{3 x^{2}}{x + 1}$

Simplify (x^4+14x^3+40x^2)/(x^2+9x+20)*(3x+15)/(x^2+11x+10)

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