Problem 6

Considering the expression $g(x) = f(x) - 2x^{2}$, we can also write the function at the negative input as $g(-x) = f(-x) - 2(-x)^{2} = f(-x) - 2x^{2}$.

Subtracting these equations, we get the following relation:
$$g(x) - g(-x) = f(x) - 2x^{2} - (f(-x) - 2x^{2}).$$

Given the initial condition $f(x) + f(-x) = 4x^{2} + 2$, we can substitute $f(-x)$ from this equation to obtain:
$$g(x) - g(-x) = f(x) - 2x^{2} - (4x^{2} + 2 - f(x)).$$
Simplifying, we find that:
$$g(x) - g(-x) = -2.$$

But since we are interested in the sum of $g(x)$ and $g(-x)$, we can add the equation $g(x) - g(-x) = -2$ to $2g(x)$ to eliminate $g(-x)$, which gives us:
$$2g(x) = 2 + 2g(x),$$
and after rearranging, we get:
$$g(x) + g(-x) = 2.$$

This implies that the graph of $g(x)$ is symmetric about the point $(0, 1)$, and therefore, the points at which $g(x)$ attains its maximum and minimum values are also symmetric about $(0, 1)$. Consequently, the sum of the maximum and minimum values of $g(x)$, $M + m$, must be equal to 2 times the $y$-coordinate of the point of symmetry:

$$\boxed{M + m = 2}.$$

Therefore, the correct answer is B: $2$.