Sure, let's go through the solution step-by-step:
1. Identify the components of the expression:
The given expression is:
[tex]\[
\frac{2 \times 3 - 3x^2 + 4x}{2x^2 - 3}
\][/tex]
2. Simplify the numerator:
The numerator is:
[tex]\[
2 \times 3 - 3x^2 + 4x
\][/tex]
Simplifying [tex]\( 2 \times 3 \)[/tex], we get:
[tex]\[
2 \times 3 = 6
\][/tex]
So, the numerator becomes:
[tex]\[
6 - 3x^2 + 4x
\][/tex]
3. Write the complete fraction:
Now, substituting back into the fraction, we have:
[tex]\[
\frac{6 - 3x^2 + 4x}{2x^2 - 3}
\][/tex]
4. Simplify the expression:
To simplify this expression, we observe that there are no common factors between the numerator and denominator that can be factored out directly. Thus, the simplified form of the given fraction retains the same form:
[tex]\[
\frac{6 - 3x^2 + 4x}{2x^2 - 3}
\][/tex]
Therefore, the detailed components and the final simplified expression are:
- Numerator: [tex]\(6 - 3x^2 + 4x\)[/tex]
- Denominator: [tex]\(2x^2 - 3\)[/tex]
- Simplified Expression: [tex]\(\frac{6 - 3x^2 + 4x}{2x^2 - 3}\)[/tex]
This reflects directly in the result provided:
[tex]\[
\left(-3x^2 + 4x + 6, 2x^2 - 3, \frac{-3x^2 + 4x + 6}{2x^2 - 3}\right)
\][/tex]
