What must be subtracted from 4x⁴ - 2x³ - 6x² + x - 5 so that the result is exactly divisible by 2x² + 3x - 2 ?

Since 2x² + 3x - 2 is of degree 2, so when 
p(x) = 4x⁴ - 2x³ - 6x² + x - 5 is divided by
q(x) = 2x² + 3x - 2, the remainder should be a linear expression (degree of remainder < degree of divisor).
Let the remainder r(x) = ax + b for exact division this remainder should be subtracted from p(x) Now let 
  f(x) = p(x) - r(x)
        = 4x⁴ - 2x³ - 6x² + x - 5 - (ax + b)
  f(x) = 4x⁴ - 2x³ - 6x² + (1 - a) x - 5 - b
Again, we have
q(x) = 2x² + 3x - 2
         = 2x² + 4x - x - 2
         = 2x (x + 2) - 1(x + 2)
        = (x + 2) (2x - 1)
Since x + 2 and 2x - 1 are factors of q(x) and f(x) is exactly divisible by q(x), hence (x + 2) and (2x - 1) are also factors of f(x), i.e.
f(-2) = 0 and f( ) = 0
.^.. f(-2)=4(- 2)⁴ - 2(- 2)³ - 6(- 2)² + (1 - a) (- 2) - b - 5 = 0
⇒ 64 + 16 - 24 - 2 + 2a - b - 5 = 0
⇒  2a - b = - 49 …… (i)
Again f( ) = 0
⇒ 4 ()⁴- 2( )³ - 6 ()² + (1 - a) x  - b - 5 = 0
⇒ - - + - -b - 5 = 0
⇒ - - b - 6 = 0
⇒ a + 2b = - 12 …… (ii)
Solving eqns (i) and (ii), we get a = - 22 and b = 5.
.^.. r(x) = - 22x + 5 should be subtracted.