1. **State the problem:** Simplify the expression $ (4 + x)^2 - (3 - x)(3 + x) $. 2. **Recall formulas:** - Square of a binomial: $ (a + b)^2 = a^2 + 2ab + b^2 $ - Difference of squares: $ (a - b)(a + b) = a^2 - b^2 $ 3. **Apply formulas:** - Expand $ (4 + x)^2 = 4^2 + 2 \cdot 4 \cdot x + x^2 = 16 + 8x + x^2 $ - Expand $ (3 - x)(3 + x) = 3^2 - x^2 = 9 - x^2 $ 4. **Substitute back:** $$ (4 + x)^2 - (3 - x)(3 + x) = (16 + 8x + x^2) - (9 - x^2) $$ 5. **Simplify:** $$ 16 + 8x + x^2 - 9 + x^2 = (16 - 9) + 8x + (x^2 + x^2) = 7 + 8x + 2x^2 $$ 6. **Final answer:** $$ 2x^2 + 8x + 7 $$