To solve the equation:

\[
4 + \frac{3}{u - 2} = \frac{-5}{u - 3}
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, multiply both sides of the equation by the common denominator \((u - 2)(u - 3)\).

\[
(u - 2)(u - 3) \left( 4 + \frac{3}{u - 2} \right) = (u - 2)(u - 3) \cdot \frac{-5}{u - 3}
\]

Distribute and simplify:

\[
(u - 2)(u - 3) \cdot 4 + 3(u - 3) = -5(u - 2)
\]

This expands as follows:

\[
4(u - 2)(u - 3) + 3(u - 3) = -5(u - 2)
\]

### Step 2: Expand both sides
First, expand the left-hand side:

\[
4(u^2 - 5u + 6) + 3(u - 3) = -5(u - 2)
\]

Expanding each part:

\[
4u^2 - 20u + 24 + 3u - 9 = -5u + 10
\]

Simplify:

\[
4u^2 - 17u + 15 = -5u + 10
\]

### Step 3: Move all terms to one side
Move all terms to the left-hand side:

\[
4u^2 - 17u + 15 + 5u - 10 = 0
\]

Simplify:

\[
4u^2 - 12u + 5 = 0
\]

### Step 4: Solve the quadratic equation
Now, solve the quadratic equation \(4u^2 - 12u + 5 = 0\) using the quadratic formula:

\[
u = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(5)}}{2(4)}
\]

Simplify:

\[
u = \frac{12 \pm \sqrt{144 - 80}}{8}
\]
\[
u = \frac{12 \pm \sqrt{64}}{8}
\]
\[
u = \frac{12 \pm 8}{8}
\]

So, we have two solutions:

\[
u = \frac{12 + 8}{8} = \frac{20}{8} = 2.5
\]
\[
u = \frac{12 - 8}{8} = \frac{4}{8} = 0.5
\]

### Step 5: Check for extraneous solutions
We must check whether either solution makes any denominator zero. 

- If \(u = 2.5\), neither \(u - 2 = 0.5\) nor \(u - 3 = -0.5\) is zero, so \(u = 2.5\) is valid.
- If \(u = 0.5\), neither \(u - 2 = -1.5\) nor \(u - 3 = -2.5\) is zero, so \(u = 0.5\) is also valid.

### Final Answer:
The solutions are:

\[
u = 2.5 \quad \text{and} \quad u = 0.5
\]

Would you like more details on any of these steps, or do you have any other questions?

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Here are 5 related questions to further your understanding:
1. How do you determine the common denominator in a rational equation?
2. What are the conditions under which a solution to a rational equation might be extraneous?
3. Can you solve quadratic equations using factoring instead of the quadratic formula?
4. What is the discriminant, and how does it determine the number of solutions to a quadratic equation?
5. How would the solution change if the quadratic had complex roots?

**Tip:** Always check for extraneous solutions when solving equations with rational expressions, especially those that involve fractions!

This step-by-step solution explains how to solve the rational equation 4 + 3/(u - 2) = -5/(u - 3). It includes using common denominators, expanding the equation, and applying the quadratic formula. Suitable for students in Grades 9-11.

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