The binomial theorem states that for any integer $n$:

$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

In this case, $a = 2x$, $b = -3$, and $n = 6$. Expanding $(2x - 3)^6$ using the binomial theorem:

$(2x - 3)^6 = \sum_{k=0}^6 \binom{6}{k} (2x)^{6-k} (-3)^k$

Step-by-Step Expansion:

1. General term: $T_k = \binom{6}{k} (2x)^{6-k} (-3)^k$

2. Individual terms:

   • For $k = 0$: $\binom{6}{0}(2x)^6(-3)^0 = 1 \cdot (64x^6) \cdot 1 = 64x^6$
   • For $k = 1$: $\binom{6}{1}(2x)^5(-3)^1 = 6 \cdot (32x^5) \cdot (-3) = -576x^5$
   • For $k = 2$: $\binom{6}{2}(2x)^4(-3)^2 = 15 \cdot (16x^4) \cdot 9 = 2160x^4$
   • For $k = 3$: $\binom{6}{3}(2x)^3(-3)^3 = 20 \cdot (8x^3) \cdot (-27) = -4320x^3$
   • For $k = 4$: (\binom{6}{4}(2x)^2(-3)^4 = 15 \cdot (