To factor the polynomial \(12x^2 + x - 6\) completely, let's follow a systematic approach.
### Step 1: Identification of the polynomial
We have the polynomial given as:
[tex]\[ 12x^2 + x - 6 \][/tex]
### Step 2: Finding the factors
We need to express this polynomial as a product of two binomials of the form \((ax + b)(cx + d)\).
### Step 3: Verifying the given choices
We need to determine which among the given options will result in the original polynomial when multiplied out.
Option A: \((3x - 2)(4x + 3)\)
[tex]\[
\begin{aligned}
(3x - 2)(4x + 3) & = 3x \cdot 4x + 3x \cdot 3 - 2 \cdot 4x - 2 \cdot 3 \\
& = 12x^2 + 9x - 8x - 6 \\
& = 12x^2 + x - 6
\end{aligned}
\][/tex]
This indeed matches our original polynomial.
Option B: \((12x - 2)(x + 3)\)
[tex]\[
\begin{aligned}
(12x - 2)(x + 3) & = 12x \cdot x + 12x \cdot 3 - 2 \cdot x - 2 \cdot 3 \\
& = 12x^2 + 36x - 2x - 6 \\
& = 12x^2 + 34x - 6
\end{aligned}
\][/tex]
This does not match our original polynomial.
Option C: \((4x - 2)(2x + 3)\)
[tex]\[
\begin{aligned}
(4x - 2)(2x + 3) & = 4x \cdot 2x + 4x \cdot 3 - 2 \cdot 2x - 2 \cdot 3 \\
& = 8x^2 + 12x - 4x - 6 \\
& = 8x^2 + 8x - 6
\end{aligned}
\][/tex]
This does not match our original polynomial.
Option D: \((12x - 3)(x + 2)\)
[tex]\[
\begin{aligned}
(12x - 3)(x + 2) & = 12x \cdot x + 12x \cdot 2 - 3 \cdot x - 3 \cdot 2 \\
& = 12x^2 + 24x - 3x - 6 \\
& = 12x^2 + 21x - 6
\end{aligned}
\][/tex]
This also does not match our original polynomial.
### Conclusion
Of the given options, only Option A:
[tex]\[
(3x - 2)(4x + 3)
\][/tex]
when multiplied, gives us the original polynomial \(12x^2 + x - 6\).
Therefore, the completely factored form of the polynomial \(12x^2 + x - 6\) is:
[tex]\[ (3x - 2)(4x + 3) \][/tex]
Correspondingly, the correct answer is choice:
[tex]\[ \boxed{A} \][/tex]
