Simplify ((x^2+6x+5)/(x^2+8x+15)*(x^3-3x^2-10x)/(x^2+3x+2))÷((x^2-4x-5)/(x^2+4x+3))

$\frac{x^{2} + 6 x + 5}{x^{2} + 8 x + 15} \cdot \frac{x^{3} - 3 x^{2} - 10 x}{x^{2} + 3 x + 2} \div \frac{x^{2} - 4 x - 5}{x^{2} + 4 x + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 6 x + 5}{x^{2} + 8 x + 15} \cdot \frac{x^{3} - 3 x^{2} - 10 x}{x^{2} + 3 x + 2} \frac{x^{2} + 4 x + 3}{x^{2} - 4 x - 5}$

Step 2

Factor $x^{2} + 6 x + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Write the factored form using these integers.

$\frac{(x + 1) (x + 5)}{x^{2} + 8 x + 15} \cdot \frac{x^{3} - 3 x^{2} - 10 x}{x^{2} + 3 x + 2} \frac{x^{2} + 4 x + 3}{x^{2} - 4 x - 5}$

Step 3

Factor $x^{2} + 8 x + 15$ 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 $15$ and whose sum is $8$ .

$3, 5$

Write the factored form using these integers.

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

Step 4

Simplify the numerator.

Factor $x$ out of $x^{3} - 3 x^{2} - 10 x$ .

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

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

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

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

Factor $x$ out of $- 10 x$ .

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

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

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

Factor $x$ out of $x (x^{2} - 3 x) + x \cdot - 10$ .

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

Factor $x^{2} - 3 x - 10$ 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 $- 3$ .

$- 5, 2$

Write the factored form using these integers.

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

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

Step 5

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

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

Step 6

Simplify terms.

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x + 5}{(x + 3) (x + 5)}$ by $\frac{x (x - 5) (x + 2)}{x + 2}$ .

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

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Factor $x^{2} + 4 x + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Step 8

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Write the factored form using these integers.

$\frac{x (x - 5)}{x + 3} \cdot \frac{(x + 1) (x + 3)}{(x - 5) (x + 1)}$

Step 9

Simplify terms.

Cancel the common factor of $x - 5$ .

Factor $x - 5$ out of $x (x - 5)$ .

$\frac{(x - 5) x}{x + 3} \cdot \frac{(x + 1) (x + 3)}{(x - 5) (x + 1)}$

Cancel the common factor.

$\frac{(x - 5) x}{x + 3} \cdot \frac{(x + 1) (x + 3)}{(x - 5) (x + 1)}$

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $(x + 1) (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x \frac{x + 1}{x + 1}$

Combine $x$ and $\frac{x + 1}{x + 1}$ .

$\frac{x (x + 1)}{x + 1}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{x (x + 1)}{x + 1}$

Divide $x$ by $1$ .

$x$

Simplify ((x^2+6x+5)/(x^2+8x+15)*(x^3-3x^2-10x)/(x^2+3x+2))÷((x^2-4x-5)/(x^2+4x+3))

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