Question

If ax³+ax²+bx²+x-6 has x+2 as a factor and leaves remainder when divided by x-2, find a and b.

Answer 1

There is some data missing.  the value of reminder when divided by x-2.

P(x) = a x³ + a x² + b x² + x - 6
Reminder of P(x) when divided by x+2 is = P(-2)  that is zero  as x+2 is a factor of P(x).

P(-2) = -8 a + 4 a + 4 b -2 - 6 = -4a +4b - 8 = 0

So  b-a = 2    or,    b = a+2       --- equation 1

When divided by x-2, the reminder is P(2) = 8 a + 4a+4b+2-6 = 12a+4b - 4
   = 12 a + 4 a + 8 - 4 =  16 a + 4      this is not zero as per question.
     so  a ≠ -4/16 or -1/4

As per the data in the question, b - a = 2   and a≠ -1/4

 Other data perhaps is missing.