1. **State the problem:** Given the function $f(x) = x^2 + 2x + 9$, find $f(x+2)$, $f(x+10)$, and $f(x+h)$ where $h$ is any constant. 2. **Recall the function substitution rule:** To find $f(a)$ for some expression $a$, replace every $x$ in the original function with $a$. 3. **Find $f(x+2)$:** $$f(x+2) = (x+2)^2 + 2(x+2) + 9$$ Expand and simplify: $$(x+2)^2 = x^2 + 4x + 4$$ $$2(x+2) = 2x + 4$$ So, $$f(x+2) = x^2 + 4x + 4 + 2x + 4 + 9 = x^2 + 6x + 17$$ 4. **Find $f(x+10)$:** $$f(x+10) = (x+10)^2 + 2(x+10) + 9$$ Expand and simplify: $$(x+10)^2 = x^2 + 20x + 100$$ $$2(x+10) = 2x + 20$$ So, $$f(x+10) = x^2 + 20x + 100 + 2x + 20 + 9 = x^2 + 22x + 129$$ 5. **Find $f(x+h)$ where $h$ is any constant:** $$f(x+h) = (x+h)^2 + 2(x+h) + 9$$ Expand and simplify: $$(x+h)^2 = x^2 + 2hx + h^2$$ $$2(x+h) = 2x + 2h$$ So, $$f(x+h) = x^2 + 2hx + h^2 + 2x + 2h + 9 = x^2 + (2h + 2)x + (h^2 + 2h + 9)$$ **Final answers:** $$f(x+2) = x^2 + 6x + 17$$ $$f(x+10) = x^2 + 22x + 129$$ $$f(x+h) = x^2 + (2h + 2)x + (h^2 + 2h + 9)$$