Dividing polynomial fractions with varying term quantities

I'm working through an old algebra book as a refresher and I've come across what should be a simple polynomial division. The exercise prompts the reader to perform the following operation:

$$\frac{{a}^{2}-9}{{a}^{2}+3a}\xf7\frac{a-3}{4}$$

I started off by inverting the $\xf7$ sign by instead multiplying by the reciprocal which results in:

$$\frac{{a}^{2}-9}{{a}^{2}+3a}\cdot \frac{4}{a-3}$$

I spent nearly 2 hours at this point trying everything my mind could conjure in terms of factoring, simplifying, and multiplying, but none of my attempts ever arrived at the listed answer:

$$\frac{4}{a}$$

If someone could help me through the steps required to solve this, you'll have taught a man to fish.

I'm working through an old algebra book as a refresher and I've come across what should be a simple polynomial division. The exercise prompts the reader to perform the following operation:

$$\frac{{a}^{2}-9}{{a}^{2}+3a}\xf7\frac{a-3}{4}$$

I started off by inverting the $\xf7$ sign by instead multiplying by the reciprocal which results in:

$$\frac{{a}^{2}-9}{{a}^{2}+3a}\cdot \frac{4}{a-3}$$

I spent nearly 2 hours at this point trying everything my mind could conjure in terms of factoring, simplifying, and multiplying, but none of my attempts ever arrived at the listed answer:

$$\frac{4}{a}$$

If someone could help me through the steps required to solve this, you'll have taught a man to fish.