Let's perform the addition and subtraction of the two functions \( f(x) \) and \( g(x) \).

### Addition \( (f(x) + g(x)) \):

\[
f(x) + g(x) = (3x^2 - 2x + 1) + (4x^2 + 5x - 4)
\]

Combine like terms:

\[
f(x) + g(x) = (3x^2 + 4x^2) + (-2x + 5x) + (1 - 4)
\]

\[
f(x) + g(x) = 7x^2 + 3x - 3
\]

### Subtraction \( (f(x) - g(x)) \):

\[
f(x) - g(x) = (3x^2 - 2x + 1) - (4x^2 + 5x - 4)
\]

Distribute the negative sign and combine like terms:

\[
f(x) - g(x) = 3x^2 - 4x^2 - 2x - 5x + 1 + 4
\]

\[
f(x) - g(x) = -x^2 - 7x + 5
\]

### Final Results:
- **Addition**: \( f(x) + g(x) = 7x^2 + 3x - 3 \)
- **Subtraction**: \( f(x) - g(x) = -x^2 - 7x + 5 \)

Would you like any further analysis or explanation?

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Here are 8 related questions you might consider:
1. What are the roots of the resulting function from \( f(x) + g(x) = 0 \)?
2. What are the roots of the resulting function from \( f(x) - g(x) = 0 \)?
3. What is the derivative of \( f(x) + g(x) \)?
4. What is the derivative of \( f(x) - g(x) \)?
5. How do the graphs of \( f(x) + g(x) \) and \( f(x) - g(x) \) compare to the original functions?
6. What is the integral of \( f(x) + g(x) \) with respect to \( x \)?
7. What is the integral of \( f(x) - g(x) \) with respect to \( x \)?
8. Can you factor the resulting expressions \( 7x^2 + 3x - 3 \) or \( -x^2 - 7x + 5 \)?

**Tip:** When subtracting functions, it's essential to handle the sign changes carefully to avoid mistakes in combining like terms.

Explore detailed solutions for adding and subtracting functions in algebra. Learn how to combine like terms and distribute operations with polynomials. Suitable for Grades 9-12 students and anyone seeking in-depth understanding of algebraic operations.

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