Let's break down the given information and solve the problem step by step.

We have the following functions:

1. $F(x) = 2x^3 - 3x^2 + x - 6$
2. $g(x) = x^2 - 4x + 5$
3. $h(x) = x + 3$

The tasks are:

1. To find $(F + g)(x)$, which is the sum of $F(x)$ and $g(x)$.
2. To find $(g \times h)(x)$, which is the product of $g(x)$ and $h(x)$.

a. $(F + g)(x)$

We add the functions $F(x)$ and $g(x)$: $(F + g)(x) = F(x) + g(x)$ Substituting the given functions: $(F + g)(x) = (2x^3 - 3x^2 + x - 6) + (x^2 - 4x + 5)$ Now combine like terms: $(F + g)(x) = 2x^3 + (-3x^2 + x^2) + (x - 4x) + (-6 + 5)$ $(F + g)(x) = 2x^3 - 2x^2 - 3x - 1$

b. $(g \times h)(x)$

We multiply the functions $g(x)$ and $h(x)$: $(g \times h)(x) = g(x) \times h(x)$ Substituting the given functions: $(g \times h)(x) = (x^2 - 4x + 5)(x + 3)$ Now apply the distributive property (FOIL method): $(g \times h)(x) = x^2(x + 3) - 4x(x + 3) + 5(x + 3)$ Multiply each term: $(g \times h)(x) = x^3 + 3x^2 - 4x^2 - 12x + 5x + 15$ Combine like terms: $(g \times h)(x) = x^3 - x^2 - 7x + 15$

Final Results:

1. $(F + g)(x) = 2x^3 - 2x^2 - 3x - 1$
2. $(g \times h)(x) = x^3 - x^2 - 7x + 15$

Would you like a more detailed explanation for any part of the solution?

Here are some related questions to explore:

1. How do you factor a cubic polynomial like $x^3 - x^2 - 7x + 15$?
2. Can you solve $F(x) = 0$ for the roots of the cubic equation?
3. What is the domain and range of $(F + g)(x)$?
4. Can you graph the functions $F(x)$, $g(x)$, and $h(x)$ together?
5. How does the product rule for derivatives apply to functions like $g(x) \times h(x)$?

Tip: When adding or multiplying polynomials, always combine like terms carefully to avoid mistakes.