Certainly! Let's factorize the quadratic expression [tex]\( 4x^2 + 5x - 6 \)[/tex].
### Step-by-Step Solution:
1. Identify the Quadratic Expression:
The expression we need to factorize is [tex]\( 4x^2 + 5x - 6 \)[/tex].
2. Look for Factors:
We aim to write the quadratic expression in the form [tex]\((Ax + B)(Cx + D)\)[/tex], where [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] are constants that we need to determine.
3. Set up the Factoring Process:
We look for two numbers that multiply to give the product of [tex]\(4 \times -6 = -24\)[/tex], and add to give the middle coefficient, which is [tex]\(5\)[/tex].
4. Find the Two Numbers:
Considering the factors of -24, we find that [tex]\(8\)[/tex] and [tex]\(-3\)[/tex] work since [tex]\(8 \times (-3) = -24\)[/tex] and [tex]\(8 + (-3) = 5\)[/tex].
5. Rewrite the Middle Term:
Using these two numbers, we can rewrite the expression:
[tex]\[
4x^2 + 5x - 6 = 4x^2 + 8x - 3x - 6
\][/tex]
6. Factor by Grouping:
Group the terms in pairs:
[tex]\[
4x^2 + 8x - 3x - 6 = (4x^2 + 8x) + (-3x - 6)
\][/tex]
7. Factor Out the Greatest Common Factor (GCF) from Each Pair:
[tex]\[
4x(x + 2) - 3(x + 2)
\][/tex]
8. Factor Out the Common Binomial Factor:
Notice that [tex]\(x + 2\)[/tex] is a common factor:
[tex]\[
= (x + 2)(4x - 3)
\][/tex]
### Result:
The factorized form of [tex]\(4x^2 + 5x - 6\)[/tex] is:
[tex]\[
(x + 2)(4x - 3)
\][/tex]
Therefore, the factorized expression for [tex]\(4x^2 + 5x - 6\)[/tex] is [tex]\((x + 2)(4x - 3)\)[/tex].
