Problem Statement
Consider the expansion of $(2x - 3)(x + 2)^5$.
**(a)** [3 marks] Find the first four terms of $(x + 2)^5$ in descending powers of $x$.
**(b)** [3 marks] Hence, find the coefficient of $x^4$ in the expansion of $(2x - 3)(x + 2)^5$.
Verified Solution & Marking Scheme
Part (a) [M1]
Using binomial formula: $\binom{5}{0}x^5 + \binom{5}{1}x^4(2) + \binom{5}{2}x^3(2^2) + \binom{5}{3}x^2(2^3)$
Part (a) [A1]
$x^5 + 5(2)x^4 + 10(4)x^3 + 10(8)x^2 = x^5 + 10x^4 + 40x^3 + 80x^2$
Part (b) [M1]
Terms contributing to $x^4$: $(2x)(40x^3) + (-3)(10x^4)$
Part (b) [A1]
$80x^4 - 30x^4 = 50x^4$
Part (b) [A1]
Coefficient is $50$