Simplify (x^2+2x-15)/(4x)*(2x^4)/(x^3+5x^2)

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

Step 1

Factor $x^{2} + 2 x - 15$ 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 $2$ .

$- 3, 5$

Write the factored form using these integers.

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

Step 2

Simplify terms.

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

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

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

Factor $x^{2}$ out of $5 x^{2}$ .

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

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 5$ .

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

Combine.

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

Step 3

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

Multiply $x^{2}$ by $x$ .

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

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

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

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

Add $2$ and $1$ .

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

Step 4

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $(x - 3) (2 x^{4})$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $x^{4}$ and $x^{3}$ .

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

$\frac{x^{3} ((x - 3) (x))}{2 x^{3}}$

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x - 3) (x)}{2}$

Step 7

Reorder factors in $\frac{(x - 3) (x)}{2}$ .

$\frac{x (x - 3)}{2}$

Simplify (x^2+2x-15)/(4x)*(2x^4)/(x^3+5x^2)

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