Prove that a^2-1 is divisible by 8 for all odd integers a (induction)?

I am running into an issue when trying to prove the above statement using induction. My attempt is shown below (only mathematical steps and short explanation included, not formal proof):
If a in ZZ is odd, then EE l in ZZ s.t. a=2l+1, and if 8|a^2-1, EE m in ZZ s.t. 8m=a^2-1, so:

8m=(2l+1)^2-1

8m=4l^2+4l+1-1

Induction step, n=k+1

8m=4(k+1)^2+4(k+1)-1

8m=4k^2+12k+8
8m=4(k^2-1)+12k+12
8m=4(8m)+12k+12
8m=8(4m)+12(k+1)

But if I have 8m=8(4m)+8((3k)/2+3/2), the RHS !inZZ.
Any guidance would be greatly appreciated!