To find the values of a and b, we can use polynomial long division. The dividend is x^3 + ax^2 + bx + 6, the divisor is x - 2, and the remainder is 3 when divided by x - 3.

1. Polynomial Long Division:
We divide x^3 + ax^2 + bx + 6 by x - 2 using polynomial long division. The quotient will be a polynomial of degree 2, and the remainder will be a polynomial of degree 0 (a constant).

x - 2 | x^3 + ax^2 + bx + 6
We start by dividing the highest degree terms:
x times (x - 2) gives x^2 - 2x^2.
Subtracting this from the dividend, we get:
x^3 + ax^2 + bx + 6 - (x^2 - 2x^2) = -x^2 + bx + 6
Now we bring down the next term (bx):
x - 2 | -x^2 + bx + 6
-x^2 + 2x^2 = x^2
bx - 2x = (b - 2)x
Subtracting this from the remaining polynomial:
-x^2 + bx + 6 - (x^2 - 2x) = bx - 2x + 6
Bring down the last term (6):
x - 2 | bx - 2x + 6
b - 2 = b - 2
6 - 2(2) = 6 - 4 = 2

The remainder is 2, not 3 as given in the problem statement. So, it appears there might be an error in the problem or the information provided is incorrect.

2. Conclusion:
Based on the polynomial long division, we cannot find the values of a and b since the remainder obtained is 2, not 3. It is important to verify the problem statement and the given information to ensure accuracy.

What do u want answer or explanation of this question