Which of the following is equivalent to `(2 a^(2)-5a -1)/(a-3)` ?

Options

A. `2a -2`
B. `2a +1 -(2)/(a -3)`
C. `2a + (2)/(a -3)`
D. `2a +1 + (2)/(a -3)`

Answer

To solve the expression \((2a^2 - 5a - 1)/(a - 3)\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{2a^2 - 5a - 1}{a - 3} \]

### Step 2: Split the Middle Term
We need to rewrite the middle term \(-5a\) in a way that helps us factor the numerator. We can split \(-5a\) into \(-6a + a\):
\[ 2a^2 - 6a + a - 1 \]

### Step 3: Add and Subtract 2
To facilitate factoring, we can add and subtract 2 in the numerator:
\[ 2a^2 - 6a + a - 1 + 2 - 2 = 2a^2 - 6a + a + 1 - 2 \]

### Step 4: Group the Terms
Now we can group the terms:
\[ (2a^2 - 6a) + (a + 1) - 2 \]

### Step 5: Factor Out Common Terms
Now we factor out the common terms from the grouped expressions:
\[ 2a(a - 3) + 1(a - 3) - 2 \]

### Step 6: Factor Out \((a - 3)\)
We can factor out \((a - 3)\) from the first two terms:
\[ (a - 3)(2a + 1) - 2 \]

### Step 7: Substitute Back into the Expression
Now we substitute this back into our original expression:
\[ \frac{(a - 3)(2a + 1) - 2}{a - 3} \]

### Step 8: Cancel the Common Terms
Now we can cancel \((a - 3)\) from the numerator and the denominator:
\[ 2a + 1 - \frac{2}{a - 3} \]

### Final Answer
Thus, the expression simplifies to:
\[ 2a + 1 + \frac{2}{a - 3} \]