Final answer:
To find the value of k, we need to determine the value of x when the expression x - (2x-2) is a factor of f(x). By setting the expression equal to zero and solving for x, we find that x = 0 or x = 1. Substituting these values into f(x), we can find the corresponding values of k.
Step-by-step explanation:
To find the value of k, we need to determine the value of x when the expression x - (2x-2) is a factor of f(x). This means that when we substitute x - (2x-2) into f(x), it should equal zero.
So, we have (x - (2x-2))(x^3 + 3x^2 + k) = 0. Expanding this expression, we get x^4 - 2x^3+2x - 2x^2 + 4 = 0.
To make sure the above equation is equal to zero, the coefficient of the x term must be zero. Therefore, -2x^3 + 2x = 0. Solving for x, we have x = 0 or x = 1.
Substituting x = 0 into f(x), we get 0^3 + 3(0)^2 + k = 0 + 0 + k = 0, therefore, k = 0.
Substituting x = 1 into f(x), we get 1^3 + 3(1)^2 + k = 1 + 3 + k = 0, therefore, k = -4.