Answer
\[\begin{align}
& \text{a}\text{. }4{{x}^{3}}+6{{x}^{2}}+2x \\
& \text{b}\text{. }4{{x}^{3}}+6{{x}^{2}}+2x \\
\end{align}\]
Work Step by Step
\[\begin{align}
& \text{a}\text{. }\frac{d}{dx}{{\left( {{x}^{2}}+x \right)}^{2}}\text{ using the chain rule} \\
& \frac{d}{dx}{{\left( {{x}^{2}}+x \right)}^{2}}=2{{\left( {{x}^{2}}+x \right)}^{2-1}}\frac{d}{dx}\left( {{x}^{2}}+x \right) \\
& \text{ }=2\left( {{x}^{2}}+x \right)\left( 2x+1 \right) \\
& \text{Simplifying} \\
& \text{ }=2\left( 2{{x}^{3}}+{{x}^{2}}+2{{x}^{2}}+x \right) \\
& \text{ }=2\left( 2{{x}^{3}}+3{{x}^{2}}+x \right) \\
& \text{ }=4{{x}^{3}}+6{{x}^{2}}+2x \\
& \\
& \text{b}\text{. Expanding }{{\left( {{x}^{2}}+x \right)}^{2}},\text{ recall that }{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}} \\
& {{\left( {{x}^{2}}+x \right)}^{2}}={{x}^{4}}+2{{x}^{3}}+{{x}^{2}} \\
& \text{Differentiating} \\
& \frac{d}{dx}\left( {{x}^{4}}+2{{x}^{3}}+{{x}^{2}} \right)=4{{x}^{3}}+6{{x}^{2}}+2x \\
\end{align}\]
