Question

Divide Rational Expressions:$\frac{x^2\hspace{0.17em}+\hspace{0.17em}3x\hspace{0.17em}-\hspace{0.17em}10}{4x}÷\left(2x^2+\hspace{0.17em}20x\hspace{0.17em}+\hspace{0.17em}50\right)$

Short answer

The required answer is$\frac{x-2}{8x\left(x+5\right)}$.

Step-by-step solution

Step 1. Given information. 

The given expression is$\frac{x^2\hspace{0.17em}+\hspace{0.17em}3x\hspace{0.17em}-\hspace{0.17em}10}{4x}÷\left(2x^2+\hspace{0.17em}20x\hspace{0.17em}+\hspace{0.17em}50\right)$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{x^2\hspace{0.17em}+\hspace{0.17em}3x\hspace{0.17em}-\hspace{0.17em}10}{4x}÷\left(2x^2+\hspace{0.17em}20x\hspace{0.17em}+\hspace{0.17em}50\right) \\ =\frac{\frac{x^2+3x-10}{4x}}{2x^2+20x+50} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} \frac{x^2+3x-10}{4x\left(2x^2+20x+50\right)} \\ =\frac{\left(x-2\right)\left(x+5\right)}{4x\left(2x^2+20x+50\right)} \\ =\frac{\left(x-2\right)\left(x+5\right)}{4x·\hspace{0.17em}2{\left(x+5\right)}^2} \end{gathered}$$

Apply exponent law.

$$\begin{gathered} \frac{\left(x-2\right)\left(x+5\right)}{4x·\hspace{0.17em}2\left(x+5\right)\left(x+5\right)} \\ =\frac{x-2}{4x·\hspace{0.17em}2\left(x+5\right)} \\ =\frac{x-2}{8x\left(x+5\right)} \end{gathered}$$