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The equation you provided is:

\[
\frac{4}{x-2} - \frac{3}{x+1} = \frac{8}{(x-2)(x+1)}
\]

Let's solve it step by step.

### Step 1: Combine the left side of the equation
On the left side, the fractions have different denominators, so we need to find a common denominator. The common denominator for \( \frac{4}{x-2} \) and \( \frac{3}{x+1} \) is \( (x-2)(x+1) \).

Rewriting each term:

\[
\frac{4}{x-2} = \frac{4(x+1)}{(x-2)(x+1)}
\]

\[
\frac{3}{x+1} = \frac{3(x-2)}{(x-2)(x+1)}
\]

So, the equation becomes:

\[
\frac{4(x+1)}{(x-2)(x+1)} - \frac{3(x-2)}{(x-2)(x+1)} = \frac{8}{(x-2)(x+1)}
\]

### Step 2: Combine the numerators
Now that the denominators are the same, we can subtract the numerators:

\[
\frac{4(x+1) - 3(x-2)}{(x-2)(x+1)} = \frac{8}{(x-2)(x+1)}
\]

Expand the numerators:

\[
4(x+1) = 4x + 4
\]
\[
3(x-2) = 3x - 6
\]

So the equation becomes:

\[
\frac{(4x + 4) - (3x - 6)}{(x-2)(x+1)} = \frac{8}{(x-2)(x+1)}
\]

Simplifying the numerator:

\[
4x + 4 - 3x + 6 = x + 10
\]

Now we have:

\[
\frac{x + 10}{(x-2)(x+1)} = \frac{8}{(x-2)(x+1)}
\]

### Step 3: Solve the equation
Since the denominators on both sides are the same, we can set the numerators equal to each other:

\[
x + 10 = 8
\]

Solve for \(x\):

\[
x = 8 - 10 = -2
\]

### Step 4: Check for restrictions
The original denominators contain \(x - 2\) and \(x + 1\), so \(x = 2\) and \(x = -1\) would be restrictions (since they would make the denominator zero). However, \(x = -2\) is not a restriction, so it is a valid solution.

Thus, the solution to the equation is:

\[
x = -2
\]

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. How do you solve equations with fractions involving different denominators?
2. What happens if the variable cancels out in such equations?
3. Can you explain the concept of restrictions in rational equations?
4. How do you factor quadratic expressions like \((x-2)(x+1)\)?
5. What are some common mistakes to avoid when solving rational equations?

### Tip:
Always check for restrictions in rational equations by ensuring no denominator becomes zero at the solution value!

Solve the equation 4/(x-2) - 3/(x+1) = 8/((x-2)(x+1))

Learn how to solve rational equations like 4/(x-2) - 3/(x+1) = 8/((x-2)(x+1)) using factoring and common denominators. Detailed step-by-step solution ideal for high school students (Grades 9-10).

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