We are given two functions $f(x) = x + 2$ and $g(x) = 3x^2 + 6x - 8$, and we need to find $g(f(x))$.

Step-by-Step Solution:

1. First, substitute $f(x) = x + 2$ into the function $g(x)$. That is, wherever there is an $x$ in $g(x)$, we will replace it with $x + 2$.

$g(f(x)) = g(x + 2) = 3(x + 2)^2 + 6(x + 2) - 8$

2. Now expand each part of the expression:

   • Expanding $(x + 2)^2$: $(x + 2)^2 = x^2 + 4x + 4$

   • Substitute this into $g(f(x))$: $g(f(x)) = 3(x^2 + 4x + 4) + 6(x + 2) - 8$

3. Distribute the constants:

   • Distribute the 3 to each term inside the first parenthesis: $3(x^2 + 4x + 4) = 3x^2 + 12x + 12$

   • Distribute the 6 to each term inside the second parenthesis: $6(x + 2) = 6x + 12$

4. Now add all the terms together: $g(f(x)) = (3x^2 + 12x + 12) + (6x + 12) - 8$

5. Combine like terms: $g(f(x)) = 3x^2 + (12x + 6x) + (12 + 12 - 8)$ $g(f(x)) = 3x^2 + 18x + 16$

So, the polynomial expression for $g(f(x))$ is:

$g(f(x)) = 3x^2 + 18x + 16$

Would you like any further details or explanations?

Here are 5 related questions to further explore this concept:

1. How do you find $f(g(x))$ for the given functions $f(x)$ and $g(x)$?
2. What are the domain and range of the composite function $g(f(x))$?
3. Can you verify the degree of the polynomial obtained for $g(f(x))$?
4. How would the result change if $f(x) = 2x + 1$ instead of $x + 2$?
5. How can you graph $g(f(x))$ using the given polynomial?

Tip: When finding composite functions, always carefully substitute the inner function and fully simplify the expression before combining like terms.