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The given proportion is:

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

### Step-by-step Solution:

1. **Cross-multiply** to eliminate the fractions:
   
   \[
   4(b - 3) = 2(4b + 3)
   \]

2. **Distribute** both sides:

   \[
   4b - 12 = 8b + 6
   \]

3. **Move all terms involving \(b\)** to one side:

   \[
   4b - 8b = 6 + 12
   \]

   Simplifying:

   \[
   -4b = 18
   \]

4. **Solve for \(b\)**:

   \[
   b = \frac{18}{-4} = -\frac{9}{2}
   \]

Thus, the solution is:

\[
b = -\frac{9}{2}
\]

Would you like more details on any step or have further questions?

### Additional Questions:
1. What happens if both sides of the equation have the same denominator?
2. Can you verify the solution by plugging \( b = -\frac{9}{2} \) back into the original proportion?
3. What are common mistakes to avoid when cross-multiplying in proportions?
4. How does solving proportions help in solving rational equations?
5. What if the proportion had additional terms like \(\frac{4}{4b + 3} + \frac{1}{b - 3} = 0\)?

**Tip**: Always check for extraneous solutions, especially when dealing with denominators that might become zero!

Solve the proportion 4 / (4b + 3) = 2 / (b - 3)

Learn how to solve the proportion 4 / (4b + 3) = 2 / (b - 3) step by step using cross-multiplication. This problem covers key algebraic concepts and is suitable for students in Grades 8-10.

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