Question regarding local maxes and minimums?

Question

For what values of a and b does the function f(x)=x^3+ax^2+bx+2 have a local maximum when x=-3 and a local minimum when x=-1? The above equation should be read as x cubed plus ax squared plus bx plus 2. Just being safe

Answer 1

f'(x) = 3x^2 + 2ax + b
x = -3 and x = -1 are critical points, so
f'(-3) = f'(-1) = 0 or
3(-3)^2 + 2a(-3) + b = 0 and
3(-1)^2 + 2a(-1) + b = 0
Solve for a and b.
27 - 6a + b = 0
3 - 2a + b = 0
or
6a - b = 27
2a - b = 3
Subtract the second equation from the first.
4a = 24
a = 6
2(6) - b = 3
12 - b = 3
-b = -9
b = 9
Answer: a = 6 and b = 9

Answer 2

f'(x)= 3x^2+2ax+b=0, the roots of this polynomial should be -1 and -3, so 3(x+3)(x+1)=3x^2+2ax+b.
If you equal out the coefficients of each polynomial you see that 12=2a and 9=b. So there are you have the solutions: b=9 and a=6