Okay, I think I see what's happened. The first equals sign in your solution is correct: the fraction _is_ equal to (4(ST+2S²))/(4(2T²+4ST)). We can cancel the 4s by dividing both sides by 4 to get (ST+2S²)/(2T²+4ST), which is part of what you did, but you tried to cancel some variables you can't actually cancel.
Always remember that when we "cancel" on a fraction, we have to multiply or divide _everything_ on both sides by the same thing, and this is exactly the same for _algebraic_ fractions. We currently have (ST+2S²)/(2T²+4ST), but it's a bit hard to see something we can divide both sides by to make it simpler.
So, next step: try to factor the top and bottom! Notice each term on the top has an S, and each term on the bottom has a 2T, so we get S(T+2S)/2T(T+2S). Aha - this uncovers a hidden common factor. Both sides have a (T+2S) factor! So we can divide both sides by it to get S/2T.
I hope this helps - let me know if you're still confused and I'll try my best to explain.