`(3a-1)^(2)-6a+2` = _____

Options

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

Answer

To solve the expression \((3a-1)^{2}-6a+2\), we will factor it step by step.

### Step 1: Expand the square
First, we expand \((3a-1)^{2}\):
\[ (3a-1)^{2} = (3a-1)(3a-1) = 9a^{2} - 6a + 1 \]
So, we rewrite the expression:
\[ 9a^{2} - 6a + 1 - 6a + 2 \]

### Step 2: Combine like terms
Now, combine the like terms:
\[ 9a^{2} - 6a - 6a + 1 + 2 = 9a^{2} - 12a + 3 \]

### Step 3: Factor out the common factor
Next, we can factor out the common factor from the quadratic expression:
\[ 9a^{2} - 12a + 3 = 3(3a^{2} - 4a + 1) \]

### Step 4: Factor the quadratic expression
Now, we need to factor the quadratic \(3a^{2} - 4a + 1\). We look for two numbers that multiply to \(3 \times 1 = 3\) and add to \(-4\). The numbers are \(-3\) and \(-1\):
\[ 3a^{2} - 3a - 1a + 1 \]
Now, we can group the terms:
\[ = (3a^{2} - 3a) + (-1a + 1) \]
Factoring by grouping:
\[ = 3a(a - 1) - 1(a - 1) \]
Now, we can factor out \((a - 1)\):
\[ = (3a - 1)(a - 1) \]

### Final Answer
Putting it all together, we have:
\[ (3a-1)^{2}-6a+2 = 3(3a-1)(a-1) \]