Simplify (x^2+10x)/(x(x+10)^2)*((x+10)^2)/(x^2+19x+90)

$\frac{x^{2} + 10 x}{x {(x + 10)}^{2}} \cdot \frac{{(x + 10)}^{2}}{x^{2} + 19 x + 90}$

Step 1

Factor $x$ out of $x^{2} + 10 x$ .

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

$\frac{x \cdot x + 10 x}{x {(x + 10)}^{2}} \cdot \frac{{(x + 10)}^{2}}{x^{2} + 19 x + 90}$

Factor $x$ out of $10 x$ .

$\frac{x \cdot x + x \cdot 10}{x {(x + 10)}^{2}} \cdot \frac{{(x + 10)}^{2}}{x^{2} + 19 x + 90}$

Factor $x$ out of $x \cdot x + x \cdot 10$ .

$\frac{x (x + 10)}{x {(x + 10)}^{2}} \cdot \frac{{(x + 10)}^{2}}{x^{2} + 19 x + 90}$

Step 2

Factor $x^{2} + 19 x + 90$ 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 $90$ and whose sum is $19$ .

$9, 10$

Write the factored form using these integers.

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

Step 3

Combine.

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

Step 4

Multiply $x + 10$ by ${(x + 10)}^{2}$ by adding the exponents.

Move ${(x + 10)}^{2}$ .

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

Multiply ${(x + 10)}^{2}$ by $x + 10$ .

Raise $x + 10$ to the power of $1$ .

$\frac{x ({(x + 10)}^{2} {(x + 10)}^{1})}{x {(x + 10)}^{2} ((x + 9) (x + 10))}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x + 10)}^{2 + 1}}{x {(x + 10)}^{2} ((x + 9) (x + 10))}$

Add $2$ and $1$ .

$\frac{x {(x + 10)}^{3}}{x {(x + 10)}^{2} ((x + 9) (x + 10))}$

Step 5

Multiply ${(x + 10)}^{2}$ by $x + 10$ by adding the exponents.

Move $x + 10$ .

$\frac{x {(x + 10)}^{3}}{x ((x + 10) {(x + 10)}^{2}) (x + 9)}$

Multiply $x + 10$ by ${(x + 10)}^{2}$ .

Raise $x + 10$ to the power of $1$ .

$\frac{x {(x + 10)}^{3}}{x ({(x + 10)}^{1} {(x + 10)}^{2}) (x + 9)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x + 10)}^{3}}{x {(x + 10)}^{1 + 2} (x + 9)}$

Add $1$ and $2$ .

$\frac{x {(x + 10)}^{3}}{x {(x + 10)}^{3} (x + 9)}$

Step 6

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x {(x + 10)}^{3}}{x {(x + 10)}^{3} (x + 9)}$

Rewrite the expression.

$\frac{{(x + 10)}^{3}}{{(x + 10)}^{3} (x + 9)}$

Step 7

Cancel the common factor of ${(x + 10)}^{3}$ .

Cancel the common factor.

$\frac{{(x + 10)}^{3}}{{(x + 10)}^{3} (x + 9)}$

Rewrite the expression.

$\frac{1}{x + 9}$

Simplify (x^2+10x)/(x(x+10)^2)*((x+10)^2)/(x^2+19x+90)

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