Let's solve the problem step by step:

We are given:

• $f(x) = 2x^2 + 7x - 15$
• $g(x) = 3 - 2x$

We need to calculate $\frac{f(x)}{g(x)}$ for all defined values.

Step 1: Factor $f(x)$

The quadratic $f(x) = 2x^2 + 7x - 15$ needs to be factored. To factor this, find two numbers that multiply to $2 \times (-15) = -30$ and add to $7$. These numbers are $10$ and $-3$.

Now split the middle term: $f(x) = 2x^2 + 10x - 3x - 15$ Group terms: $f(x) = (2x^2 + 10x) - (3x + 15)$ Factorize each group: $f(x) = 2x(x + 5) - 3(x + 5)$ Factor out $(x + 5)$: $f(x) = (2x - 3)(x + 5)$

Step 2: Write $\frac{f(x)}{g(x)}$

Substitute the factored form of $f(x)$: $\frac{f(x)}{g(x)} = \frac{(2x - 3)(x + 5)}{3 - 2x}$

Step 3: Simplify $g(x) = 3 - 2x$

Notice that $g(x) = 3 - 2x = -(2x - 3)$. Substitute this into the fraction: $\frac{f(x)}{g(x)} = \frac{(2x - 3)(x + 5)}{-(2x - 3)}$

Cancel $2x - 3$, as long as $2x - 3 \neq 0$: $\frac{f(x)}{g(x)} = -(x + 5)$

Final Answer:

$\frac{f(x)}{g(x)} = -x - 5$ This corresponds to option (1).

Would you like a more detailed explanation of any step? Here are 5 related questions for practice:

1. How do you factorize a quadratic equation effectively?
2. What happens if $g(x) = 0$? Why do we exclude these values?
3. How do you simplify a rational expression like $\frac{f(x)}{g(x)}$?
4. Why is it important to rewrite $g(x) = 3 - 2x$ as $-(2x - 3)$?
5. What is the domain of $\frac{f(x)}{g(x)}$, and how is it determined?

Tip: Always check for restrictions in the denominator before canceling terms in a fraction!