What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$?
We simply distribute the $4$ to get $4x^4 + 12x^2 + 4.$ Then, the sum of the squares of the coefficients is $4^2 + 12^2 + 4^2 = \boxed{176}.$

Note that the constant term $4$ is indeed a coefficient: it is the coefficient of $x^0$.