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Let's solve the problem step by step.

We are given the following functions:

- \( F(x) = 2x^3 - 3x^2 + x - 6 \)
- \( g(x) = x^2 - 4x + 5 \)
- \( h(x) = x + 3 \)

### Part (a): Determine \( (F + g)(x) \)

To add the functions \( F(x) \) and \( g(x) \), we simply add the corresponding terms.

\[
(F + g)(x) = F(x) + g(x)
\]
Substitute the functions:

\[
(F + g)(x) = (2x^3 - 3x^2 + x - 6) + (x^2 - 4x + 5)
\]

Now, combine like terms:

\[
(F + g)(x) = 2x^3 + (-3x^2 + x^2) + (x - 4x) + (-6 + 5)
\]

\[
(F + g)(x) = 2x^3 - 2x^2 - 3x - 1
\]

Thus, the result is:

\[
(F + g)(x) = 2x^3 - 2x^2 - 3x - 1
\]

Would you like me to continue with part (b) \( (g \times h)(x) \), or do you have any questions about this part?

Here are some related questions to explore:
1. How do you subtract functions instead of adding them?
2. What is the degree of the resulting polynomial after adding two functions?
3. How do you find the value of \( (F + g)(2) \)?
4. How do you graph \( F(x) + g(x) \)?
5. What happens when you add a linear function to a cubic function?

Tip: When combining functions, always make sure to combine like terms carefully for accuracy.

Learn how to solve function operations by adding F(x) = 2x^3 - 3x^2 + x - 6 and g(x) = x^2 - 4x + 5, and multiplying g(x) and h(x) = x + 3. This problem covers key algebraic concepts, including polynomials and function operations. Suitable for high school students.

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