We will break the polynomial into two groups: $x^3+4x^2$ and $2x+8\text{.}$ $$\begin{aligned} x^3+4x^2+2x+8\amp=\left(x^3+4x^2\right)+(2x+8) \end{aligned}$$ and now each group has its own greatest common factor to factor out: $$\begin{aligned} \amp=x^2\highlight{(x+4)}+2\highlight{(x+4)} \end{aligned}$$ and now the binomial $(x+4)$ appears twice and can be factored out: $$\begin{aligned} \amp=\highlight{(x+4)}\left(x^2+2\right) \end{aligned}$$