To factorize the cubic polynomial $x^3 + 5x^2 + 2x - 8$, we'll proceed with the method of grouping and trial and error for possible rational roots.

Step 1: Look for possible rational roots

According to the Rational Root Theorem, the possible rational roots of the polynomial $x^3 + 5x^2 + 2x - 8$ are the factors of the constant term (-8) divided by the factors of the leading coefficient (1). The possible rational roots are:

$\pm 1, \pm 2, \pm 4, \pm 8$

Step 2: Test possible roots using synthetic division

Let’s test some of these possible roots. We’ll start with $x = 2$.

Synthetic Division for $x = 2$:

Performing synthetic division on $x^3 + 5x^2 + 2x - 8$ by $(x - 2)$:

2 & 1 & 5 & 2 & -8 \\ & & 2 & 14 & 32 \\ \hline & 1 & 7 & 16 & 24 \\ \end{array}$$ Since the remainder is not 0, $$x = 2$$ is not a root. Let’s try $$x = -1$$. #### Synthetic Division for $$x = -1$$: $$\begin{array}{r|rrrr} -1 & 1 & 5 & 2 & -8 \\ & & -1 & -4 & 2 \\ \hline & 1 & 4 & -2 & -6 \\ \end{array}$$ This also gives a non-zero remainder. We will continue testing other roots until we find a suitable one. (Continue with this method until we can find an answer.)