Answer
$\dfrac{3x+2}{(1+x)^{\frac{1}{2}}}$
Work Step by Step
The LCD is $(1+x)^{\frac{1}{2}}$.
Make the expressions similar using the LCD to obtain:
$\dfrac{x}{(1+x)^{\frac{1}{2}}}+2(1+x)^{\frac{1}{2}}\\
=\dfrac{x}{(1+x)^{\frac{1}{2}}}+2(1+x)^{\frac{1}{2}} \cdot \dfrac{(1+x)^{\frac{1}{2}}}{(1+x)^{\frac{1}{2}}}$
Use the rule $a^m \cdot a^n = a^{m+n}$, then simplify to obtain:
$=\dfrac{x}{(1+x)^{\frac{1}{2}}}+\dfrac{2(1+x)^{\frac{1}{2}+\frac{1}{2}}}{(1+x)^{\frac{1}{2}}}$
$=\dfrac{x}{(1+x)^{\frac{1}{2}}}+\dfrac{2(1+x)^{1}}{(1+x)^{\frac{1}{2}}}$
$=\dfrac{x}{(1+x)^{\frac{1}{2}}}+\dfrac{2(1+x)}{(1+x)^{\frac{1}{2}}}$
$=\dfrac{x}{(1+x)^{\frac{1}{2}}}+\dfrac{2+2x}{(1+x)^{\frac{1}{2}}}$
Add the rational expressions to obtain:
$=\dfrac{x+2+2x}{(1+x)^{\frac{1}{2}}}$
$=\dfrac{3x+2}{(1+x)^{\frac{1}{2}}}$
Hence, the correct answer is $\frac{3x+2}{(1+x)^{\frac{1}{2}}}$.
