lim_x-2 x6 -24x-16/x³+2x-12. How i can solve it by factorisation?

lim_x-2 x6 -24x-16/x³+2x-12
Substitute x=2 in x⁶-24x-16 ; Then we get
2⁶-24(2)-16=0
So we got 0. So, x=2 is one of its root. Let us divide it by synthetic division.

The quotient here is
x⁵+2x⁴+4x³+8x²+16x+8
Similarly substitute x=2 in x³+2x-12.
2³+2(2)-12=0.
So we got 0. So, x=2 is one of its root. Let us divide it by synthetic division.

The quotient here is
x²+2x+6.
Let us apply the limit now.
$$\begin{gathered} \underset{x\to 2}{\lim }\frac{x^6-24x-16}{x^3-2x+12} \\ =\underset{x\to 2}{\lim }\frac{(x-2)(x^5+2x^4+4x^3+8x^2+16x+8)}{(x-2)(x^2+2x+6)} \\ =\underset{x\to 2}{\lim }\frac{(x^5+2x^4+4x^3+8x^2+16x+8)}{(x^2+2x+6)} \\ =\frac{(2^5+2(2^4)+4(2^3)+8(2^2)+16(2)+8)}{\left(\right(2{)}^2+2\left(2\right)+6)} \\ =\frac{168}{14}=12 \end{gathered}$$