Finding k in a Polynomial Function with Extrema

Consider the polynomial function defined by the equation $$P(x) = 3x^4 - 8x^3 + 2x^2 + k$$, where $$k$$ is a constant. If the polynomial has a local maximum at $$x = 1$$ and a local minimum at $$x = -2$$, what is the value of $$k$$?

To find the local extrema, we need to first calculate the derivative $$P'(x)$$ and set it equal to zero to find critical points. The derivative is given by: $$P'(x) = 12x^3 - 24x^2 + 4x$$.