To find $g(f(x))$, we need to substitute the function $f(x)$ into the function $g(x)$.

Given:

• $f(x) = -2x - 10$
• $g(x) = x^2 + 5x + 9$

Steps:

1. First, substitute $f(x) = -2x - 10$ into $g(x)$: $g(f(x)) = g(-2x - 10)$

2. Now substitute $-2x - 10$ for $x$ in the expression for $g(x)$: $g(-2x - 10) = (-2x - 10)^2 + 5(-2x - 10) + 9$

3. Expand the terms: $(-2x - 10)^2 = (4x^2 + 40x + 100)$ $5(-2x - 10) = -10x - 50$

4. Now, substitute these expanded values back into the equation: $g(-2x - 10) = 4x^2 + 40x + 100 - 10x - 50 + 9$

5. Combine like terms: $g(-2x - 10) = 4x^2 + (40x - 10x) + (100 - 50 + 9)$ $g(-2x - 10) = 4x^2 + 30x + 59$

Thus, $g(f(x)) = 4x^2 + 30x + 59$.

Would you like further details or explanations?

Here are some related questions to explore:

1. How do you compute $g(f(x))$ for different types of functions?
2. What are the steps for finding $f(g(x))$?
3. How do transformations affect the composition of functions?
4. Can $g(f(x))$ be used to model real-world scenarios?
5. How would $g(f(x))$ change if $f(x)$ were a different linear function?

Tip: Always expand terms carefully in compositions to avoid mistakes in signs or coefficients!