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The question asks for \( g(f(x)) \), given that \( f(x) = 2x^2 + x \) and \( g(x) = 4x + 1 \).

To solve for \( g(f(x)) \):

1. Start by finding \( f(x) \):
   \[
   f(x) = 2x^2 + x
   \]
   
2. Now substitute \( f(x) \) into the function \( g(x) = 4x + 1 \). We need to replace \( x \) in \( g(x) \) with \( f(x) \):
   \[
   g(f(x)) = 4(2x^2 + x) + 1
   \]

3. Distribute the \( 4 \) to both terms inside the parentheses:
   \[
   g(f(x)) = 4(2x^2) + 4(x) + 1
   \]
   \[
   g(f(x)) = 8x^2 + 4x + 1
   \]

Thus, \( g(f(x)) = 8x^2 + 4x + 1 \).

The correct answer is:
- \( \boxed{8x^2 + 4x + 1} \)

Would you like more details on any part of the solution?

Here are five related questions to practice:

1. How do you solve for \( g(f(x)) \) when \( g(x) = x^2 - 1 \) and \( f(x) = 3x + 2 \)?
2. If \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 5 \), how would you compute \( g(f(x)) \)?
3. What is the composite function \( g(f(x)) \) if \( g(x) = \sqrt{x+2} \) and \( f(x) = x^2 - 1 \)?
4. How do you find the inverse of a composite function like \( g(f(x)) \)?
5. Can you explain how function composition is used in real-life applications, such as in physics or economics?

**Tip:** Always simplify each function before performing the composition to avoid mistakes.

If f(x) = 2x^2 + x and g(x) = 4x + 1, Find g(f(x))

Learn how to solve g(f(x)) for given functions f(x) = 2x^2 + x and g(x) = 4x + 1. This problem covers function composition and quadratic functions, suitable for Grades 10-12.

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