Which of the following expressions is equivalent to `(x^2 - 2x -5)/(x-3)` ?

Options

A. `x-5-20/(x-3)`
B. `x-5-10/(x-3)`
C. `x+1 -8/(x-3)`
D. `x+1-2/(x-3)`

Explanation

To solve the expression \((x^2 - 2x - 5)/(x - 3)\) and determine which of the given options is equivalent, we will perform polynomial long division. Here’s the step-by-step solution:

### Step 1: Set up the division
We need to divide \(x^2 - 2x - 5\) by \(x - 3\).

### Step 2: Divide the leading terms
Divide the leading term of the numerator \(x^2\) by the leading term of the denominator \(x\):
\[ \frac{x^2}{x} = x \]
This means \(x\) is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \(x\) by the entire divisor \(x - 3\):
\[ x(x - 3) = x^2 - 3x \]
Subtract this from the original polynomial:
\[ (x^2 - 2x - 5) - (x^2 - 3x) = -2x + 3x - 5 = x - 5 \]

### Step 4: Repeat the process
Now, we need to divide \(x - 5\) by \(x - 3\):
Divide the leading term \(x\) by \(x\):
\[ \frac{x}{x} = 1 \]
Now, multiply \(1\) by the divisor \(x - 3\):
\[ 1(x - 3) = x - 3 \]
Subtract this from \(x - 5\):
\[ (x - 5) - (x - 3) = -5 + 3 = -2 \]

### Step 5: Write the final result
Now, we can express the result of the division:
\[ \frac{x^2 - 2x - 5}{x - 3} = x + 1 - \frac{2}{x - 3} \]

### Conclusion
Thus, the expression \((x^2 - 2x - 5)/(x - 3)\) simplifies to:
\[ x + 1 - \frac{2}{x - 3} \]
This corresponds to **Option D**: \(x + 1 - \frac{2}{x - 3}\).