For what values of b is (x-b) a factor of 3x^3 +(b+3)x^2 -(4b^2 +b-7)x-4?

Let f(x) = 3x^3 + (b+3)x^2 - (4b^2+b-7)x-4 Then (x-b) is a factor of f(x) if and only if f(b)=0, because if (x-b) is a factor of f(x) then we can write f(x)=(x-b)g(x) for some other (quadratic) polynomial g, which when evaluated at b clearly equals zero. Hence we have that (x-b) is a factor of f, if and only if: 3b^3 + (b+3)b^2 - (4b^2+b-7)b-4=0 That is, if: 2b^2+7b-4=0 (2b-1)(b+4)=0 So b=-4 or 1/2

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