Finding Coefficients in a Polynomial

Let $$P(x)$$ be a polynomial defined as follows:

$$P(x) = 4x^4 - 3x^3 + bx^2 + cx - 12$$

Given that the polynomial $P(x)$ has a factor of $$x - 2$$, what is the value of $$b + c$$ if the polynomial can also be expressed as:

$$P(x) = (x - 2)(4x^3 + dx^2 + ex + f)$$

Use the Remainder Theorem for polynomial division to find the coefficients and determine $$b + c$$.