\[\frac{x^{2} + 4x - 8}{x - 4} =\]
\[= \frac{x^{2} - 4x + 8x - 32 + 24}{x - 4} =\]
\[= \frac{x(x - 4) + 8 \cdot (x - 4) + 24}{x - 4} =\]
\[= \frac{(x - 4)(x + 8) + 24}{x - 4} =\]
\[= \frac{(x - 4)(x + 8)}{x - 4} + \frac{24}{x - 4} =\]
\[= x + 8 + \frac{24}{x - 4}\]