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.