The Binomial Theorem

$\dfrac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$. Cancel before multiplying rather than computing $8!$.

The coefficients are row 4 of Pascal's triangle: 1, 4, 6, 4, 1. Five terms for a power of 4.

$\binom{6}{2}(2x)^2 = 15 \times 4x^2 = 60x^2$. The whole of $2x$ is squared.

The counter $r$ runs from 0 to 7, giving $n+1 = 8$ terms.

With $n = -2$: the $x$ term is $-2x$, and the $x^2$ term is $\dfrac{(-2)(-3)}{2}x^2 = 3x^2$.

$\binom{6}{4}(3)^2(-x)^4 = 15 \times 9 \times x^4 = 135x^4$. The fourth power makes the minus sign positive.