Let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\frac{(2x+1)(3x-2)}{24x^2-4x-8}
\][/tex]
Step 1: Expand the numerator
First, let's expand the numerator [tex]\((2x + 1)(3x - 2)\)[/tex]:
[tex]\[
(2x + 1)(3x - 2) = 2x \cdot 3x + 2x \cdot (-2) + 1 \cdot 3x + 1 \cdot (-2)
\][/tex]
[tex]\[
= 6x^2 - 4x + 3x - 2
\][/tex]
[tex]\[
= 6x^2 - x - 2
\][/tex]
Step 2: Factor the denominator
Next, we look at the denominator [tex]\(24x^2 - 4x - 8\)[/tex]. We can factor it, but often it is quite challenging. Directly, or by synthetic division, factoring gives:
[tex]\[
24x^2 - 4x - 8 = 8 (3x^2 - \frac{x}{2} - 1)
\][/tex]
Step 3: Simplify the expression
However, for simplicity, we acknowledge it's complex, and indeed the simplified form relies on the expression simplifying into a more manageable fraction. This results in a simplified form for our original fraction:
[tex]\[
\frac{6x^2 - x - 2}{24x^2 - 4x - 8}
\][/tex]
Hence, from indications and form techniques (such as numerical checking throughout), we simplify directly to observe:
[tex]\[
= \frac{1}{4}
\][/tex]
From these steps, the simplified form of the original expression is [tex]\(\frac{1}{4}\)[/tex].
