Let's evaluate the expressions based on the operations and the given functions:

Given Functions:

1. $g(x) = x^2 + 2x - 8$
2. $f(x) = x + 3$
3. $h(x) = \frac{x + 7}{2 - x}$
4. $p(x) = 2x - 7$
5. $t(x) = \frac{x - x}{x + 3}$
6. $v(x) = x^2 + 5x + 4$

1. $f(x) \cdot t(x)$

$f(x) \cdot t(x) = (x + 3) \cdot \frac{x - x}{x + 3} = (x + 3) \cdot 0 = 0$ Since $t(x) = \frac{0}{x + 3} = 0$, the result of this operation is $0$.

2. $v(x) - h(x)$

$v(x) - h(x) = \left(x^2 + 5x + 4\right) - \frac{x + 7}{2 - x}$ This expression can be rewritten as: $v(x) - h(x) = x^2 + 5x + 4 - \frac{x + 7}{2 - x}$ You would need to find a common denominator to combine these, but this is the general form of the difference.

3. $p(x) \cdot h(x)$

$p(x) \cdot h(x) = (2x - 7) \cdot \frac{x + 7}{2 - x}$ This becomes: $p(x) \cdot h(x) = \frac{(2x - 7)(x + 7)}{2 - x}$ You can expand the numerator using the distributive property: $(2x - 7)(x + 7) = 2x^2 + 14x - 7x - 49 = 2x^2 + 7x - 49$ Thus, $p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}$

4. $g(x) - t(x)$

Since $t(x) = 0$, $g(x) - t(x) = g(x) - 0 = g(x) = x^2 + 2x - 8$

5. $f(x) \cdot t(x)$

As we saw in the first calculation, $f(x) \cdot t(x) = 0$.

6. $h(x) \div v(x)$

$\frac{h(x)}{v(x)} = \frac{\frac{x + 7}{2 - x}}{x^2 + 5x + 4}$ This can be rewritten as: $\frac{x + 7}{(2 - x)(x^2 + 5x + 4)}$ You could simplify further, depending on the specific values of $x$.

7. $p(x) \cdot h(x)$

This was already evaluated in step 3: $p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}$

Summary of Results:

1. $f(x) \cdot t(x) = 0$
2. $v(x) - h(x) = x^2 + 5x + 4 - \frac{x + 7}{2 - x}$
3. $p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}$
4. $g(x) - t(x) = x^2 + 2x - 8$
5. $f(x) \cdot t(x) = 0$
6. $\frac{h(x)}{v(x)} = \frac{x + 7}{(2 - x)(x^2 + 5x + 4)}$
7. $p(x) \cdot h(x) = \frac{2x^2 + 7x - 49}{2 - x}$

Do you have any questions or would you like further details on any step?

Related Questions:

1. What happens when you simplify $v(x) - h(x)$?
2. How does the form of $p(x) \cdot h(x)$ change when you simplify further?
3. What is the behavior of these functions for different values of $x$?
4. How do you find the domain restrictions for $h(x)$ and $v(x)$?
5. What would $f(x) \div t(x)$ look like if $t(x)$ were not zero?

Tip:

When dealing with fractions or rational expressions, always check for possible restrictions on the domain, especially when a denominator can be zero.