Factor this polynomial completely.12x2 + x – 6A.(4x – 2)(2x + 3)B.(3x – 2)(4x + 3)C.(12x – 2)(x + 3)D.(12x – 3)(x + 2)

To factor the polynomial $12x^2 + x - 6$ completely, follow these steps:

1. Identify the coefficients: The polynomial is in the form $ax^2 + bx + c$, where $a = 12$, $b = 1$, and $c = -6$.

2. Find two numbers that multiply to $a \cdot c$ and add to $b$:

   • $a \cdot c = 12 \cdot (-6) = -72$
   • We need two numbers that multiply to $-72$ and add to $1$.

   After checking possible pairs, we find that $9$ and $-8$ work because:

   • $9 \cdot (-8) = -72$
   • $9 + (-8) = 1$

3. Rewrite the middle term using these numbers: $12x^2 + 9x - 8x - 6$

4. Factor by grouping:

   • Group the terms: $(12x^2 + 9x) + (-8x - 6)$
   • Factor out the greatest common factor (GCF) from each group: $3x(4x + 3) - 2(4x + 3)$

5. Factor out the common binomial factor: $(3x - 2)(4x + 3)$

So, the polynomial $12x^2 + x - 6$ factors completely as $(3x - 2)(4x + 3)$.

Therefore, the correct answer is: B. $(3x - 2)(4x + 3)$