The problem you provided asks to solve the following, given that $F(x) = x^2 + 5x + 6$ and $H(x) = x + 2$:

1. Find $(F \cdot H)(x)$
2. Find $(F \div H)(x)$
3. Find $(F + H)(x)$

Let's solve each part step by step.

1. $(F \cdot H)(x)$

To find $(F \cdot H)(x)$, multiply the functions $F(x)$ and $H(x)$:

$(F \cdot H)(x) = F(x) \cdot H(x) = (x^2 + 5x + 6) \cdot (x + 2)$

Expand the expression:

$(F \cdot H)(x) = x^2 \cdot x + x^2 \cdot 2 + 5x \cdot x + 5x \cdot 2 + 6 \cdot x + 6 \cdot 2$

Simplifying:

$(F \cdot H)(x) = x^3 + 2x^2 + 5x^2 + 10x + 6x + 12 = x^3 + 7x^2 + 16x + 12$

So:

$(F \cdot H)(x) = x^3 + 7x^2 + 16x + 12$

2. $(F \div H)(x)$

To find $(F \div H)(x)$, divide $F(x)$ by $H(x)$:

$(F \div H)(x) = \frac{F(x)}{H(x)} = \frac{x^2 + 5x + 6}{x + 2}$

Factor the numerator:

$F(x) = x^2 + 5x + 6 = (x + 2)(x + 3)$

Now, divide:

$(F \div H)(x) = \frac{(x + 2)(x + 3)}{x + 2}$

Since $x + 2$ cancels out, we have:

$(F \div H)(x) = x + 3$

3. $(F + H)(x)$

To find $(F + H)(x)$, add the functions $F(x)$ and $H(x)$:

$(F + H)(x) = F(x) + H(x) = (x^2 + 5x + 6) + (x + 2)$

Combine like terms:

$(F + H)(x) = x^2 + 6x + 8$

Summary of Results

1. $(F \cdot H)(x) = x^3 + 7x^2 + 16x + 12$
2. $(F \div H)(x) = x + 3$
3. $(F + H)(x) = x^2 + 6x + 8$

Would you like any further details or have any questions? Here are five related questions to expand your understanding:

1. What would be $(F - H)(x)$ given the same functions?
2. How can the division $(F \div H)(x)$ change if $H(x)$ were different?
3. What is the derivative of $(F \cdot H)(x)$?
4. What would be the roots of $(F + H)(x)$?
5. How does the degree of the polynomial change in $(F \cdot H)(x)$ compared to $F(x)$ and $H(x)$ individually?

Tip: When working with polynomial expressions, always consider factoring and simplifying to make your calculations easier.