To simplify the expression, I need to distribute the 4 and the -7 to each term inside the parentheses, and then combine any like terms. So I have $4x^2-8x+8-7x^3+21x-7$. I can see that there are two terms that have an $x$ in them, so I can combine them: $4x^2-8x+8-7x^3+21x-7 = 4x^2+13x+1-7x^3$. Now I can reorder the terms by decreasing degree of $x$: $-7x^3+4x^2+13x+1$. This is the fully simplified expression. To find the sum of the squares of the coefficients, I need to square each coefficient and add them up. The coefficients are -7, 4, 13, and 1. Their squares are 49, 16, 169, and 1. Adding them up, I get 49+16+169+1 = 235. 
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?