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We are given the functions:

\[
f(x) = 5x + 1, \quad g(x) = 5x - 1, \quad \text{and} \quad h(x) = 10x^2 + 1
\]

We are asked to find \( h(x) - g(x) \).

### Step-by-step solution:

1. First, write out the expressions for \( h(x) \) and \( g(x) \):
   \[
   h(x) = 10x^2 + 1
   \]
   \[
   g(x) = 5x - 1
   \]

2. Now, subtract \( g(x) \) from \( h(x) \):
   \[
   h(x) - g(x) = (10x^2 + 1) - (5x - 1)
   \]

3. Distribute the negative sign:
   \[
   h(x) - g(x) = 10x^2 + 1 - 5x + 1
   \]

4. Simplify the expression by combining like terms:
   \[
   h(x) - g(x) = 10x^2 - 5x + 2
   \]

So, the result is:

\[
h(x) - g(x) = 10x^2 - 5x + 2
\]

However, this doesn't match any of the given options exactly, but the closest option to our result is \( 10x^2 + 8x + 2 \). There seems to be a small discrepancy in the provided options.

Would you like any more details or have any further questions?

### Here are 5 related questions to expand your understanding:
1. What is the result of \( f(x) - g(x) \) for the given functions?
2. Can you find \( f(x) \times g(x) \) and simplify the expression?
3. What is \( h(x) + g(x) \)?
4. How would you calculate the derivative of \( h(x) - g(x) \)?
5. What is the value of \( h(x) - g(x) \) when \( x = 2 \)?

**Tip**: Always distribute the negative sign carefully when subtracting functions to avoid sign errors.

Let f(x) = 5x + 1, g(x) = 5x - 1, and h(x) = 10x^2 + 1. Find h(x) - g(x).

Learn how to subtract functions with a step-by-step solution for h(x) - g(x) where h(x) = 10x^2 + 1 and g(x) = 5x - 1. This problem covers algebraic operations on quadratic and linear functions, ideal for Grades 9-11 students.

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