jmc

Note that $(x+1{)}^2\ge 0$ for all $x.$ For the remaining part of the expression, we can build a sign chart.

$$\begin{array}{cccc}& x<0& 0<x<7& 7<x\\ x& -& +& +\\ x-7& -& -& +\\ \frac{x(x+1{)}^2}{x-7}& +& -& +\end{array}$$Also, $\frac{x(x+1{)}^2}{x-7}=0$ at $x=0$ and $x=-1.$ Thus, the solution is $x\in \boxed{\{-1\}\cup [0,7)}.$