Factorles:
` 12a ^(2) b- 9ab^(2) + 6ab`

Options

A. `3 ab (4a + 3b +2)`
B. `3 ab (4a - 3b -2)`
C. `3 ab (4a - 3b +2)`
D. `3 ab (4a + 3b -2)`

Answer

To factor the expression \( 12a^2b - 9ab^2 + 6ab \), we will follow these steps:

### Step 1: Identify the common factors in each term
The given expression is \( 12a^2b - 9ab^2 + 6ab \). We will look for the greatest common factor (GCF) of the coefficients and the variables in each term.

- The coefficients are 12, -9, and 6. The GCF of these numbers is 3.
- The variable part includes \( a^2, ab^2, \) and \( ab \). The common variable factor is \( ab \) (the lowest power of \( a \) is \( a^1 \) and the lowest power of \( b \) is \( b^1 \)).

Thus, the GCF of the entire expression is \( 3ab \).

### Step 2: Factor out the GCF
Now we will factor out \( 3ab \) from each term in the expression:

\[ 12a^2b - 9ab^2 + 6ab = 3ab(4a) - 3ab(3b) + 3ab(2) \]

### Step 3: Write the expression in factored form
Now, we can write the expression as:

\[ 3ab(4a - 3b + 2) \]

### Final Answer
The factored form of the expression \( 12a^2b - 9ab^2 + 6ab \) is:

\[ 3ab(4a - 3b + 2) \]

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