How to find the coefficient of $x^2$ in $(3-2x) \left(1 + \frac{x}{2} \right)^n$

Question

In the expansion of $(3-2x) \left(1 + \frac{x}{2} \right)^n$, the coefficient of $x$ is $7$. Find the value of the constant $n$ and hence find the coefficient of $x^2$.

I have no idea how to begin to solve this.

Any help is hugely appreciated, thanks.

This question is from the A level Cambridge syllabus (9709/w16/p12).

Answer 1

Assuming n to be positive, the only way you can "make" an x term is by multiplying a constant with an x term, or vice versa. What I mean by this is, 3 times some $x^1$ term will yield an x term. Furthermore, $(-2x)$ times the constant ($x^0$) term of the expansion will also yield an x term. Note that all other terms in the expansion will yield only higher powers of x. So you can calculate the term containing x in terms of n and then set it equal to 7. Find n that way. I hope the rest is intuitive :)