About $(x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16$

Question

Studying polynomials I got the follows: $$ (x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16 $$ I can't understand from where we got this $-8x^3$. I got to simplify this polynomial just to: $$ 2x^5 + 16 $$ Can someone help me understand from where we got the expression $-8x^3$ ?

Answer 1

When expanding $(x^3-4)^2$, you get $x^6-4x^3-4x^3+16=x^6-8x^3+16.$ Then, the $x^6$ is subtracted off, and you are left with $2x^5-8x^3+16$, as desired.

Answer 2

$$\begin{align} (x^3 - 4)^2 - x^6 + 2x^5 &= x^3\cdot x^3 + (-4)^2 + 2\cdot (-4) \cdot x^3 -x^6+ 2x^5\\ &=x^6-x^6 + 16 -8x^3 + 2x^5 = 2x^5 -8x^3 + 16 \end{align} $$