Exponents and Polynomials
Factoring a quadratic by the ac-method
Factor by grouping (sometimes called the ac-method).
\[
12 x^{2}+5 x-2
\]

First, choose a form with appropriate signs.
Then, fill in the blanks with numbers to be used for grouping Finally, show the factorization.
Form:
$12 x^{2}+\square x+\square x-2$
$12 x^{2}+\square x-\square x-2$
$12 x^{2}-\square x+\square x-2$
$12 x^{2}-\square x-\square x-2$
Factorization:
Explanation
Check

Step 1 ：Multiply the coefficient of \(x^2\) (which is 12) and the constant term (which is -2) to get -24.

Step 2 ：Find two numbers that multiply to -24 and add to 5. The numbers that satisfy these conditions are 8 and -3.

Step 3 ：Rewrite the middle term of the quadratic equation (5x) as the sum of 8x and -3x to get \(12x^2 + 8x - 3x - 2\).

Step 4 ：Group the terms to get \(4x(3x + 2) - 1(3x + 2)\).

Step 5 ：Factor out the common binomial term to get \((3x + 2)(4x - 1)\).

Step 6 ：Check: Expand \((3x + 2)(4x - 1)\) to get \(12x^2 + 5x - 2\), which is the original quadratic equation. So, the factorization is correct.

Step 7 ：\(\boxed{(3x + 2)(4x - 1)}\) is the factorization of the quadratic equation \(12x^2 + 5x - 2\).