How to express 4x^2 + 2 in the form a(x + 5) + b(2x - 3^2) + c - 2.?

T = a(x + 5) + b(2x – 3x^2) + c - 2
The above assumes you meant 3x^2, not 3^2; and rearranging T
T = (-3b)x^2 + (a + 2b)x + (5a + c – 2), …. compare coefficients with
T = 4x^2 + (0)x + 2
b = -4/3
a = -2b = 8/3
c = 4 - 5a = (12 – 40)/3 = -28/3

Not sure this is possible without an x^2 component in the second form
a or b would need an x factor or c an x^2
also if c has an x factor we may be screwed too
I would assume a, b, c need to be real numbers
I'll try both ways

4x^2 + 2

a(x + 5) + b(2x - 3x^2) + c - 2

b = -4/3 (to get you a 4x^2)
a = 8/3 (eliminates -8x/3 from b; since final has 0x)
2 = c + 40/3 (from a) - 2
c = -28/3

so, now how could you set this up

ax + 5a + 2bx - 9b + c - 2 = 4x^2 + 2
ax + 2bx + 5a - 9b + c = 4x^2 + 4

good luck from there; I may be way off but I suspect a typo in the question
it appears to me there could be any number of solutions
I do not see a way to isolate any of the components
4x^2 - what combination make this
0x - what combination makes this
4 - what combination makes this