Find all values of $a$ for which the inequality $(x-a-2)(x^2-(a^2+5a-3)x+5a^3-2a^2-5a+2) \leq 0$ has at most one positive solution

You're on the right track but you've missed a case. The inequality is of the form $$(x-x_1)(x-x_2)(x-x_3)\le 0 $$ Suppose the order of the roots is $x_i\le x_j\le x_k$. Then the solutions to the inequality are $(-\infty,x_i]\cup[x_j,x_k]$. We want this set to contain at most $1$ positive number. There are two possibilities: 1) $x_k\le 0$ or 2) $x_j=x_k>0,x_i\le 0$. Option 1 implies all the roots are $\le 0$ which turns out to be impossible for the specific $x_1,x_2,x_3$. As for option 2, you have to consider the 3 cases you mentioned. That'd give you all possible values of $a$.