We are tasked with finding $g(f(x))$, where:

$f(x) = x^2 + x - 6$ $g(x) = 4x + 9$

To solve this, we need to substitute $f(x)$ into the function $g(x)$. In other words, we will replace $x$ in $g(x)$ with $f(x)$:

$g(f(x)) = 4(f(x)) + 9$

Substitute $f(x) = x^2 + x - 6$ into this expression:

$g(f(x)) = 4(x^2 + x - 6) + 9$

Now, distribute the 4:

$g(f(x)) = 4x^2 + 4x - 24 + 9$

Simplify the constants:

$g(f(x)) = 4x^2 + 4x - 15$

So, the answer is:

$g(f(x)) = 4x^2 + 4x - 15$

Summary:

$g(f(x)) = 4x^2 + 4x - 15$

Would you like more details or clarification? Here are some related questions you might find interesting:

1. How do you perform function composition in general?
2. What is the significance of function composition in real-life applications?
3. How would the result change if we had a different function for $g(x)$?
4. What is the domain of the composed function $g(f(x))$?
5. How do you graph a composition of functions like $g(f(x))$?

Tip: Always remember to substitute carefully when working with function compositions, especially with polynomials—take extra caution when expanding!