Given that (x-2) and (x+2) are factors of x^3+ax^2+bx-12. Find the values of a and b.

Given that (x-2) and (x+2) are factors of x^3+ax^2+bx-12. Find the values of a and b.

We did this in class a couple of weeks ago but I think I've lost my notes, please explain

thank you

Let f(x) = x³ + ax² + bx - 12
Remember the factor theorem:
x - 2 is a factor so f(2) = 0.
x + 2 is a factor so f(-2) = 0
8 + 4a + 2b - 12 = 0
-8 + 4a - 2b - 12 = 0
Simplify these.
4a + 2b = 4
4a - 2b = 20
Divide by 2.
2a + b = 2
2a - b = 10
Add them.
4a = 12
a = 3
Substitute into 2a + b = 2.
6 + b = 2
b = -4
Solution: a = 3, b = -4.

then the other factor must be [ x + 3]

x^3 + a x² + b x - 12 = [ x - 2 ] [ x + 2 ] [ x - 3 ] , evaluate when x = ± 1 and solve the system in a & b

a is 5 and b is 7