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We are tasked with finding \( (h \circ g)(x) \), which represents the composition of functions \( h(x) \) and \( g(x) \), where:

- \( g(x) = 2x^2 - 3x + 1 \)
- \( h(x) = 4x + 1 \)

### Step-by-step solution:
To compute \( (h \circ g)(x) \), we substitute \( g(x) \) into \( h(x) \).

1. First, substitute \( g(x) = 2x^2 - 3x + 1 \) into \( h(x) \):
   \[
   h(g(x)) = h(2x^2 - 3x + 1)
   \]
2. Since \( h(x) = 4x + 1 \), replace the \( x \) in \( h(x) \) with \( g(x) \):
   \[
   h(2x^2 - 3x + 1) = 4(2x^2 - 3x + 1) + 1
   \]
3. Distribute the \( 4 \) inside the parentheses:
   \[
   4(2x^2 - 3x + 1) = 8x^2 - 12x + 4
   \]
4. Add the \( 1 \) from the \( +1 \) at the end of \( h(x) \):
   \[
   8x^2 - 12x + 4 + 1 = 8x^2 - 12x + 5
   \]

Thus, \( (h \circ g)(x) = 8x^2 - 12x + 5 \).

### Conclusion:
The correct answer is \( 8x^2 - 12x + 5 \), which corresponds to the third option.

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Would you like further clarification on any part of the process?

Here are 5 additional questions for further practice:
1. What is the result of \( (f \circ g)(x) \) using the given functions?
2. How do you compute the composition \( (g \circ f)(x) \)?
3. What is the result of \( (h \circ f)(x) \)?
4. What does \( (k \circ h)(x) \) yield based on the functions provided?
5. Can you find \( (f \circ k)(x) \)?

**Tip:** When working with function composition, always substitute the inner function into the outer function carefully, paying attention to all operations.

Given: f(x) = 3x - 7, g(x) = 2x^2 - 3x + 1, h(x) = 4x + 1, k(x) = -x^2 + 3. Find: (h ∘ g)(x)

Explore the process of calculating the function composition (h ∘ g)(x) using polynomial functions. This example uses functions g(x) = 2x^2 - 3x + 1 and h(x) = 4x + 1, leading to a result of 8x^2 - 12x + 5. This lesson is ideal for high school students learning about algebra and function operations.

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