When the expression $4(x^2-2x+2)-7(x^3-3x+1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?
First we distribute the constants in $4(x^2-2x+2)-7(x^3-3x+1)$ to get $4x^2-8x+8-7x^3+21x-7.$ Combining like terms, we find that this is $-7x^3+4x^2+13x+1.$ Then, the sum of the squares of all the coefficients is $(-7)^2 + (4)^2 + (13)^2 + (1)^2 = 49 + 16 + 169 + 1 = \boxed{235}.$