Maths help: Second derivative differentiation

The question I am stuck on is "The equation of a curve has the form of y=e^x^2(ax^2+b) where a and b are non-zero constants. It is given that d2y/dx2 can be expressed in the form of e^x^2(cx^4+d) where c and d are non-zero constants. Prove that 5a+2b=0"
So far I have managed to differentiate the first expression twice to get , d2y/dx2=e^x^2(4ax^4+4bx^2+10ax^2+2b+2a), however I have no clue where I am meant to go to from here. I have looked at the mark scheme, and it just jumps straight to the conclusion that 5a+2b=0 so I am not sure if it just something really obvious that I am missing? (It says "equate x^2 e^x^2 = 0" also in the mark scheme before it reaches the end if that helps although I have no idea what this means.)

The x^2 terms in
(4ax^4+4bx^2+10ax^2+2b+2a)
Must be zero for it to have that form, so the coeffs must satisfy ...

Uh I am not sure what you mean. If x^2 terms are zero then we just get 2b+2a? Also why must it be 0?

What are the quadratic tetms? There are 2 in your answer, but none in the desired form, so they must disappear.
The coefficients must satisfy what for them to be identically zero?

Sorry but I am still really confused, this sounds really cryptic to me lol. So we get rid of all the x^2 terms so that we are left with just a's and b's, is that what you mean?

You have a quartic expression.
You know/want the quadratic terms in it to be zero so
4bx^2+10ax^2 = 0
What a and b relationship ensures this constraint is satisfied?

So you would want 4b+10a to equal 0 so that you can cancel out the x^2 terms. But I still don't understand why just the quadratic part needs to equal 0, why not the whole expression?

It's given that you want the expression to be a quartic with no quadratic term.

Oh yeah I didn't notice that. Thank you

I’m still confused for how to find the second derivative. I can’t get the answer in the form it’s looking for. I checked the mark scheme and can’t make sense of it