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To solve the given problem, let's break it down:

### Problem Interpretation
We are given:
- \( g(f(x)) = 4x^2 + 8x + 1 \)
- \( f(x) = 2x + 1 \)
  
We need to find \( g(x) \).

### Solution
1. Since \( g(f(x)) = 4x^2 + 8x + 1 \) and \( f(x) = 2x + 1 \), substitute \( f(x) \) with \( 2x + 1 \) in the expression for \( g \):
   \[
   g(2x + 1) = 4x^2 + 8x + 1
   \]
   
2. Let \( u = 2x + 1 \), so we need to express \( g(u) \) in terms of \( u \).

3. Solve for \( x \) in terms of \( u \):
   \[
   u = 2x + 1 \Rightarrow x = \frac{u - 1}{2}
   \]

4. Substitute \( x = \frac{u - 1}{2} \) into \( 4x^2 + 8x + 1 \):
   \[
   g(u) = 4\left(\frac{u - 1}{2}\right)^2 + 8\left(\frac{u - 1}{2}\right) + 1
   \]

5. Simplify this expression:
   - Compute \( \left(\frac{u - 1}{2}\right)^2 = \frac{(u - 1)^2}{4} = \frac{u^2 - 2u + 1}{4} \)
   - So,
     \[
     4\left(\frac{u - 1}{2}\right)^2 = u^2 - 2u + 1
     \]
   - And,
     \[
     8\left(\frac{u - 1}{2}\right) = 4(u - 1) = 4u - 4
     \]
   - Therefore

Given g(f(x)) = 4x^2 + 8x + 1 and f(x) = 2x + 1, determine g(x).

Learn how to determine g(x) from the function composition g(f(x)) = 4x^2 + 8x + 1 and f(x) = 2x + 1. This step-by-step solution demonstrates key algebraic concepts such as function composition and solving quadratic functions. Suitable for students in Grades 9-11.

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