(ASAP PLEASE) so I know the answer is 8 root three but I always come to a dead end up while working it out.

(2 + √3)^2 - (2 - √3)^2 (expand the expressions)
4 + 4√3 + 3 - (4 - 4√3 + 3) (calculate)
4 + 4√3 + 3 - (7 - 4√3) (remove the parantheses)
4 + 4√3 + 3 - 7 + 4√3 (calculate and collect like terms)
0 + 8√3 (remove 0)
8√3