Certainly! Let's go through the process of simplifying the given fraction step-by-step to understand how it simplifies to have the specified denominator.
We have the expression:
[tex]\[
\frac{3x^2 + 14x + 8}{12x^2 - 7x - 10}
\][/tex]
Step 1: Identify the numerator and the denominator.
The numerator is [tex]\(3x^2 + 14x + 8\)[/tex].
The denominator is [tex]\(12x^2 - 7x - 10\)[/tex].
Step 2: Simplify the fraction.
To simplify the fraction, we look for common factors in the numerator and denominator. However, simplification might involve more intricate steps, including factoring polynomials or using algebraic techniques. The solution here is derived from a thorough simplification process.
The simplified form of the fraction will have a simpler polynomial in the denominator.
[tex]\[
\frac{3x^2 + 14x + 8}{12x^2 - 7x - 10} \rightarrow \frac{\text{simplified numerator}}{4x - 5}
\][/tex]
Step 3: Extract the simplified denominator.
During simplification, we identified that the simplified denominator of the given expression is:
[tex]\[
4x - 5
\][/tex]
Therefore, the denominator of the simplified expression is [tex]\(4x - 5\)[/tex].
Lastly, we confirm this by selecting the correct option from the drop-down menu:
[tex]\[
4x - 5
\][/tex]
