If is a linear transformation such that

and then

Let me explain my thinking first:

so a(1+2x)+b(3+5x) = -2 - 3x

I solved and got that a = 1 ; b = -1

I then did:

1(2-3x) + -1(2-3x) = 0.

The homework system I'm using only allowed me one try on the problem, and so I want to make sure that I did it correctly, however, my gut tells me 0 is not the right answer. Is my instinct right or am I doubting myself?

Any help would be greatly appreciated, I only have one attempt and I'd like to get it right!