Let's solve the given rational division problem step by step.

The expression is:

$\dfrac{\dfrac{x^2-3x-10}{x-3}}{\dfrac{x+2}{3x-9}}$

Step 1: Simplifying the Expression

First, rewrite the division as multiplication:

$\left(\dfrac{x^2-3x-10}{x-3}\right) \times \left(\dfrac{3x-9}{x+2}\right)$

Step 2: Factor where possible

• Factor $x^2 - 3x - 10$. To factor this quadratic, we find two numbers that multiply to $-10$ and add to $-3$, which are $-5$ and $2$:

$x^2 - 3x - 10 = (x - 5)(x + 2)$

• Also, notice that $3x - 9$ can be factored as:

$3x - 9 = 3(x - 3)$

So, the expression becomes:

$\left(\dfrac{(x - 5)(x + 2)}{x - 3}\right) \times \left(\dfrac{3(x - 3)}{x + 2}\right)$

Step 3: Cancel common factors

Now, cancel out common factors between the numerator and the denominator:

• The $x - 3$ terms cancel out.
• The $x + 2$ terms also cancel out.

This leaves:

$3(x - 5)$

Thus, the quotient in lowest terms is:

$3(x - 5)$

Step 4: Determine values to exclude from the domain

We need to exclude values of $x$ that would make any denominator in the original expression equal to zero.

Denominators in the expression:

1. $x - 3 = 0$ → $x = 3$
2. $x + 2 = 0$ → $x = -2$
3. $3x - 9 = 0$ → $x = 3$

So, the values we must exclude are $x = 3$ and $x = -2$.

Final answers:

1. The quotient in lowest terms is: $3(x - 5)$

2. The values of $x$ that we must exclude are:

• $x = -2$ (Choice A)
• $x = 3$ (Choice C)

Therefore, the correct answers are Choice A and Choice C.

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Do you need further details or have any questions?

Here are 5 related questions to consider:

1. How would the solution change if the original expression included an additional factor in the denominator?
2. What happens if you encounter higher-degree polynomials in the numerator and denominator?
3. How can you simplify an expression where no common factors exist between the numerator and denominator?
4. What additional restrictions on the domain might occur if we were dealing with inequalities rather than equations?
5. How do the excluded values affect the graph of a rational function?

Tip: Always factor both the numerator and denominator fully before simplifying a rational expression to avoid missing common factors.