#### Answer

The solution is $(-1,\infty)$
The graph is:

#### Work Step by Step

$(x+3)^{2}(x+1)\gt0$
Begin immediately by finding the intervals, because all nonzero terms are on one side of the inequality and the nonzero side is given in factored form.
The factors are $x+1$ and $(x+3)^{2}$. Set them equal to $0$ and solve for $x$:
$x+1=0$
$x=-1$
$(x+3)^{2}=0$
$x+3=0$
$x=-3$
The factors are zero when $x=-1,-3$. These two numbers divide the real line into the following intervals:
$(-\infty,-3)$ $,$ $(-3,-1)$ $,$ $(-1,\infty)$
Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the the attached image below)
It can be seen from the diagram that the inequality is satisfied only on the interval $(-1,\infty)$. Also, the inequality involves $\gt$ so the endpoints don't satisfy the inequality.
The solution is $(-1,\infty)$