What is/are the solution(s) to the equation `(x)/(x+1)+(8)/(x-2)=(3)/((x+1)(x-2))`?

Options

A. `-5` only
B. `-1 and -5`
C. `1 and 5`
D. No real solution

Explanation

To solve the equation

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

we will follow these steps:

### Step 1: Find a common denominator
The left-hand side has two fractions with different denominators. The common denominator for both fractions is \((x+1)(x-2)\). We will rewrite each fraction with this common denominator.

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

Now, substituting these into the equation gives:

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

### Step 2: Eliminate the denominators
Since both sides of the equation have the same denominator, we can multiply both sides by \((x+1)(x-2)\) to eliminate the denominators:

\[ x(x-2) + 8(x+1) = 3 \]

### Step 3: Expand and simplify
Now, we will expand the left-hand side:

\[ x^2 - 2x + 8x + 8 = 3 \]

Combining like terms gives:

\[ x^2 + 6x + 8 = 3 \]

### Step 4: Rearrange the equation
Next, we will move all terms to one side of the equation:

\[ x^2 + 6x + 8 - 3 = 0 \]
\[ x^2 + 6x + 5 = 0 \]

### Step 5: Factor the quadratic equation
Now, we will factor the quadratic equation:

\[ (x + 5)(x + 1) = 0 \]

### Step 6: Solve for \(x\)
Setting each factor equal to zero gives us the potential solutions:

1. \(x + 5 = 0 \Rightarrow x = -5\)
2. \(x + 1 = 0 \Rightarrow x = -1\)

### Step 7: Check for extraneous solutions
We must check if these solutions make any denominator zero in the original equation:

- For \(x = -5\):
- Denominators: \(x + 1 = -5 + 1 = -4\) (not zero)
- \(x - 2 = -5 - 2 = -7\) (not zero)

- For \(x = -1\):
- Denominators: \(x + 1 = -1 + 1 = 0\) (zero, undefined)

Since \(x = -1\) makes the denominator zero, it is not a valid solution.

### Final Solution
Thus, the only solution to the equation is

\[ \boxed{-5}. \]

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