Very fun puzzle! Got a little stuck in the bottom left corner, but got through it.

I recently got Heat: Pedal to the Metal & really really want to get it to the table.

👀

For some rational solutions - we can quickly find that we need a=n/c and b=m/d for some integers n and m. Then, after some algebra, we need either c=d or c=(n/m)d (and the latter case forces a=b). So, for example, the following works: a=2/5, b=2/5, c=15/2, d=25/2 (here, n=3 and m=5). Fun problem!

For some irrational solutions, a=b=c=d=sqrt(2) yields 2x^2+4x+2. Similarly, any situation with a=b=sqrt(n) and c=d=sqrt(m) will work so long as nm is a square.