#### Answer

The solution is $(-\infty,-2)\cup(-2,4)$
The graph is:

#### Work Step by Step

$(x-4)(x+2)^{2}\lt0$
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 $x-4$ and $(x+2)^{2}$. Set them equal to $0$ and solve for $x$:
$x-4=0$
$x=4$
$(x+2)^{2}=0$
$x+2=0$
$x=-2$
The factors are zero when $x=4,-2$. These two numbers divide the real line into the following intervals:
$(-\infty,-2)$ $,$ $(-2,4)$ $,$ $(4,\infty)$
Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the attached image below)
It can be seen from the diagram that the inequality is satisfied only on the intervals $(-\infty,-2)$ and $(-2,4)$. Also, the inequality involves $\lt$ so the endpoints don't satisfy the inequality.
The solution is $(-\infty,-2)\cup(-2,4)$