Find the real values of $\mathit{x}$ and $\mathit{y}$ for which the following equation: $\sqrt{x^{2}+3x+8} + \left ( x+4 \right )\mathit{i} = \mathit{y}\left (2 + \mathit{i} \right )$

Question

This is the third part of the multi-part question Q4.