Answer
$$\frac{1}{3} a^{3}-\frac{1}{2} a^{2}+\frac{5}{6}$$
Work Step by Step
\begin{aligned}
\int_{-a}^{1}\left(x^{2}+x\right) d x &=\int_{-a}^{0}\left(x^{2}+x\right) d x+\int_{0}^{1}\left(x^{2}+x\right) d x\\
&=\int_{0}^{1}\left(x^{2}+x\right) d x-\int_{0}^{-a}\left(x^{2}+x\right) d x \\
&=\left(\frac{1}{3} \cdot 1^{3}+\frac{1}{2} \cdot 1^{2}\right)-\left(\frac{1}{3}(-a)^{3}+\frac{1}{2}(-a)^{2}\right)\\
&=\frac{1}{3} a^{3}-\frac{1}{2} a^{2}+\frac{5}{6}
\end{aligned}
