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It looks like you have several math problems here. I'll go through each one.

### Problem 1 (p² - 3p - 10) / (p² + p - 2):
This appears to be a rational expression. You can simplify it by factoring both the numerator and the denominator.

**Numerator**: p² - 3p - 10
- To factor: find two numbers that multiply to -10 and add up to -3. These numbers are -5 and 2.
- So, p² - 3p - 10 = (p - 5)(p + 2).

**Denominator**: p² + p - 2
- To factor: find two numbers that multiply to -2 and add up to 1. These numbers are 2 and -1.
- So, p² + p - 2 = (p + 2)(p - 1).

Now the expression becomes:

\[
\frac{(p - 5)(p + 2)}{(p + 2)(p - 1)}.
\]

Cancel out the common factor of (p + 2):

\[
\frac{p - 5}{p - 1}.
\]

### Problem 2 (x - 3) / (3 - x) = -1:
You need to simplify this equation. Notice that the denominator can be rewritten to make it easier to solve:

\[
3 - x = -(x - 3),
\]

So, the equation becomes:

\[
\frac{x - 3}{-(x - 3)} = -1.
\]

Simplifying the fraction:

\[
-1 = -1.
\]

This is true, so the equation holds for all values of x except where the denominator equals zero, i.e., \( x \neq 3 \).

### Problem 3 (3x) / (x - 3) ÷ (2x(x - 3)):
This is a division of two fractions. You can convert it to a multiplication problem by flipping the second fraction:

\[
\frac{3x}{x - 3} \times \frac{x - 3}{2x(x - 3)}.
\]

Now, cancel out the common factors of (x - 3) and x:

\[
\frac{3}{2(x - 3)}.
\]

This is the simplified form of the expression.

### Problem 4 (18x) / (2x³ - 12x² + 18x):
Factor the denominator first:

\[
2x³ - 12x² + 18x = 2x(x² - 6x + 9) = 2x(x - 3)².
\]

Now the expression becomes:

\[
\frac{18x}{2x(x - 3)²}.
\]

Cancel out the common factors of 2x:

\[
\frac{9}{(x - 3)²}.
\]

Let me know if you'd like more details or have other questions! 

Here are a few related questions to expand on:

1. How do you factor trinomials with negative middle terms like p² - 3p - 10?
2. How do you solve rational equations involving variables in the denominator?
3. Can you explain why division of fractions involves multiplying by the reciprocal?
4. What are other methods to simplify complex rational expressions?
5. How do you handle cubic terms in polynomials for factoring?

**Tip**: When dividing rational expressions, remember to multiply by the reciprocal of the second fraction instead of dividing directly!

Learn how to simplify rational expressions, solve equations, and work with fractions in algebra. This guide covers factoring, canceling common factors, and solving rational equations. Suitable for students in Grades 9-11.

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