If y=a+bx+cx^2 is a solution of d²y/dx² +2(dy/dx)+4y=4x², I have tried to find the values of a,b and c by taking the derivative and second derivative of the function the subbing these into the differential equation however I can't seem to cancel out much to get to a single constant to use to sub into the equation to solve the other two constants, can anyone suggest what I can do after subbing in values the derivatives of y into the DE to solve for the constants a,b and c? Is this the right approach to finding the values of a,b and c?

You're on the right track - not sure where your error is.

If y = cx^2+bx+a, then:

dy/dx = 2cx+b
d^2y/dx^2 = 2c

Sub into the original DE:
2c + 2(2cx+b) + 4 (cx^2+bx+a) = 4x^2

Regroup:

(4c-4)x^2 + (4c+4b)x + 2c+2b+4a = 0

From the x^2 term you have c = 1, which means that b = -1, and a = 0.

So you have y = x^2-x

Sub into the original DE and make sure it works.

Thanks I tried to group the 2c+2b+4a together and also tried working this out simultaneously, but from how u worked this out makes so much more sense. And yep subbing back into the DE works :)