Answer
The division shows a remainder of zero; thus $g(x)$ is indeed a factor of $f(x)$
Work Step by Step
$\space$$\space$$\space$$\space$$\space$$\space$$\space$$\space$$\space\space$ ${x^3-4x^2+8x-1}$
$x+2$$\space)\overline{x^4-2x^3+0x^2+15x-2}$
$\space\space\space\space\space\space\space\space\space\space\space\space\underline{x^4+2x^3}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-4x^3+0x^2$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{-4x^3-8x^2}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space8x^2+15x$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{8x^2+16x}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-x-2$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{-x-2}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space0$
Check answer:
$(x^3-4x^2+8x-1)\cdot(x+2)+0$
$x^4+2x^3-4x^3-8x^2+8x^2+16x-x-2$
$x^4-2x^3+15x-2\checkmark$
