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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.

Find g(f(x)) where f(x) = x + 2 and g(x) = 3x^2 + 6x - 8.

Learn how to find the composite function g(f(x)) where f(x) = x + 2 and g(x) = 3x^2 + 6x - 8. This step-by-step guide walks through substituting and simplifying to express the result as a polynomial.

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