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Let's solve the subtraction problem step by step.

The expression is:

\[
\frac{4x}{x+4} - \frac{5x}{x-2}
\]

### Step 1: Find the common denominator.
The denominators are \( x+4 \) and \( x-2 \). The least common denominator (LCD) is the product of these two denominators, so:

\[
\text{LCD} = (x+4)(x-2)
\]

### Step 2: Rewrite the fractions with the LCD as the denominator.
For the first fraction, multiply both the numerator and denominator by \( (x-2) \):

\[
\frac{4x}{x+4} = \frac{4x(x-2)}{(x+4)(x-2)} = \frac{4x(x-2)}{(x+4)(x-2)}
\]

For the second fraction, multiply both the numerator and denominator by \( (x+4) \):

\[
\frac{5x}{x-2} = \frac{5x(x+4)}{(x-2)(x+4)} = \frac{5x(x+4)}{(x-2)(x+4)}
\]

### Step 3: Subtract the two fractions.
Now that the denominators are the same, subtract the numerators:

\[
\frac{4x(x-2) - 5x(x+4)}{(x+4)(x-2)}
\]

### Step 4: Expand the numerators.
Expand each term in the numerator:

\[
4x(x-2) = 4x^2 - 8x
\]
\[
5x(x+4) = 5x^2 + 20x
\]

Substitute these back into the expression:

\[
\frac{(4x^2 - 8x) - (5x^2 + 20x)}{(x+4)(x-2)}
\]

### Step 5: Simplify the numerator.
Distribute the negative sign and combine like terms:

\[
(4x^2 - 8x) - (5x^2 + 20x) = 4x^2 - 8x - 5x^2 - 20x = -x^2 - 28x
\]

So the expression becomes:

\[
\frac{-x^2 - 28x}{(x+4)(x-2)}
\]

### Final Answer:
The simplified form of the subtraction is:

\[
\frac{-(x^2 + 28x)}{(x+4)(x-2)}
\]

You can leave it like this, or factor the numerator further:

\[
\frac{-x(x + 28)}{(x+4)(x-2)}
\]

Let me know if you'd like further explanation on any step, or have any questions!

---

Here are 5 related questions to deepen your understanding:
1. How do you find the least common denominator for rational expressions?
2. What is the importance of factoring in simplifying algebraic fractions?
3. Can you show how to solve similar problems with three terms in the numerator?
4. What happens when you subtract rational expressions with the same denominator?
5. Can you describe how this problem changes if one denominator was a quadratic expression?

**Tip:** Always simplify the numerator fully before performing the final division! This reduces errors.

Subtract: (4x / (x+4)) - (5x / (x-2)). Simplify your answer as much as possible.

Learn how to subtract rational expressions in algebra with a detailed step-by-step solution. This problem involves finding a common denominator and simplifying the result, focusing on core algebraic concepts. Suitable for students in Grades 9-12.

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