I like to do tables to determine signs. It follows from Bolzano's Theorem the sign will be kept between the roots. You already factored this and obtained $x=2,x=-2,x=0$.
$\begin{array}{|c|c|c|c|} & (-\infty,-2) & (-2,0)& (0,2)&(2,+\infty)\\ \hline x-2 & - &- & -&+\\ \hline x+2 &- &+ &+ &+\\ \hline x & - & - &+ &+\\ \hline p(x)& -&+&-&+\\ \hline \end{array}$
Explanation: The top row is the real line divided into the intervals by the roots we have. The first three rows of $+/-$ account for the sign of the factors in each interval, which we obtain by inspection. Finally, the sign of $p$ is obtained by "multiplying" each of the values obtained, so $-\times-\times-=-$, $-\times+\times +=+$,$\&c$.
Try and do the same for $p(x)={x^3} + 4{x^2} + 3x - 2$.