Answer
The answer is $4x^2 + 16x + 55 + \frac{222}{x - 4}$.
Work Step by Step
Now, the dividend $4x^3 - 9x + 2$ can be rewritten as $4x^3 + 0x^2 - 9x + 2$.
$\space \space \space \space \space \space \space \space \space \space \space \space 4x^2 + 16x + 55$
$x - 4 /\overline{4x^3 + 0x^2 - 9x + 2}$
$\space \space \space \space \space \space \space \space \space \space \space \space \underline{4x^3 - 16x^2}$
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 16x^2 - 9x$
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \underline{16x^2 -64x}$
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 55x + 2$
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \underline{55x - 220}$
$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 222$
The answer is $4x^2 + 16x + 55 + \frac{222}{x - 4}$.