Exponents and Polynomials
Factoring a quadratic by the ac-method

Factor by grouping (sometimes called the ac-method).
\[
4 x^{2}-5 x-6
\]

First, choose a form with appropriate signs.
Then, fill in the blanks with numbers to be used for grouping. Finally, show the factorization.

Step 1 ：Given the quadratic equation \(4x^{2}-5x-6\).

Step 2 ：The equation is in the form of \(ax^{2} + bx + c\).

Step 3 ：We use the ac-method of factoring which involves finding two numbers m and n such that they multiply to \(a*c\) (product of coefficient of \(x^{2}\) and constant term) and add up to b (coefficient of x).

Step 4 ：In this case, a = 4, b = -5, and c = -6. So we need to find two numbers m and n such that they multiply to \(4*-6 = -24\) and add up to -5.

Step 5 ：We find these two numbers to be -2 and 3, because \(-2*3 = -6\) (which is the product of a and c) and \(-2 + 3 = 1\) (which is the coefficient of x).

Step 6 ：We rewrite the middle term (bx) of the quadratic equation as the sum of (mx) and (nx). Then we can factor by grouping.

Step 7 ：The factored form of the equation is \((x - 2)*(4x + 3)\).

Step 8 ：\(\boxed{(x - 2)(4x + 3)}\) is the factored form of the given quadratic equation \(4x^{2}-5x-6\).