Stuck on quadratic inequalities question

'A gardener has been asked to lay a patio 3m longer than its width. Each slab is a square of side 0.5m and costs £4. The gardener has been asked to spend more than £220 but less than £448. Find the two possible amounts he could spend.'

What I've done: Each slab costs £4. Divided each amount by 4 to give 55 slabs and 112 slabs. Each slab is 0.25 m sqd. So 55 slabs is 13.75 m sqd and 112 slabs is 28 m sqd. x(x+3) = x sqd + 3x.
x sqd + 3x > 13.75
x sqd + 3x < 28
But they don't factorise, and formula gives long decimals, so don't think this is the route to solving...
Would really appreciate help here.

Try doing it in terms of the number of slabs along the width instead of the width. If n is the number of slabs, then n(n+6) are used and the cost is 4n(n+6). You'll find the numbers drop out much nicer this way.

Thanks Gregorius. In fact, I'd tried that as well, and did find that they factorised nicely.
x sqd + 6x > 55
x sqd + 6x < 112
I get x values of -11 and 5, and -14 and 8.
I rejected the negative values as that would give a negative number of slabs.

OK so far...

So we're left with x= 5 slabs, or 8 slabs.

I can see that n must be larger than 5 and less than 8. BUT...how do I then get to the two values of n that will give me the two different expenditures the question asks for? The answers give two exact amounts rather than inequalities - £288 and £364.

n is an integer larger than 5 and less than 8 - therefore n = 6 or n = 7. Substitute back into 4n(n+6) and you get £288 & £364.