1. **State the problem:** Given two functions $f(x) = -x^2 + 4x - 10$ and $g(x) = 4x^2 - 7x + 6$, find the functions $(f+g)(x)$ and $(f-g)(x)$. 2. **Recall the formulas:** - The sum of two functions is defined as $(f+g)(x) = f(x) + g(x)$. - The difference of two functions is defined as $(f-g)(x) = f(x) - g(x)$. 3. **Calculate $(f+g)(x)$:** $$ (f+g)(x) = (-x^2 + 4x - 10) + (4x^2 - 7x + 6) $$ Combine like terms: $$ = (-x^2 + 4x^2) + (4x - 7x) + (-10 + 6) $$ $$ = 3x^2 - 3x - 4 $$ 4. **Calculate $(f-g)(x)$:** $$ (f-g)(x) = (-x^2 + 4x - 10) - (4x^2 - 7x + 6) $$ Distribute the minus sign: $$ = -x^2 + 4x - 10 - 4x^2 + 7x - 6 $$ Combine like terms: $$ = (-x^2 - 4x^2) + (4x + 7x) + (-10 - 6) $$ $$ = -5x^2 + 11x - 16 $$ **Final answers:** $$(f+g)(x) = 3x^2 - 3x - 4$$ $$(f-g)(x) = -5x^2 + 11x - 16$$