Let's analyze the given division and identify the term containing an error in the quotient.
We start with the given expression:
[tex]\[
\frac{3x^4 + 6x^3 + 5x - 8}{x + 4}
\][/tex]
Performing polynomial long division on the numerator [tex]\(3x^4 + 6x^3 + 5x - 8\)[/tex] by the denominator [tex]\(x + 4\)[/tex], we find the quotient and the remainder.
The quotient obtained from this division is:
[tex]\[
3x^3 - 6x^2 + 24x - 91
\][/tex]
The remainder obtained is:
[tex]\[
356
\][/tex]
Given the quotient in the problem,
[tex]\[
3x^3 - 6x^2 + 29x - 91 + \frac{356}{x + 4}
\][/tex]
we need to compare this with the calculated quotient:
[tex]\[
3x^3 - 6x^2 + 24x - 91
\][/tex]
By comparing the corresponding terms in the given quotient and the calculated quotient, we notice a discrepancy in the coefficient of the [tex]\(x\)[/tex] term:
- The coefficient of [tex]\(x\)[/tex] in the given quotient is [tex]\(29\)[/tex].
- The coefficient of [tex]\(x\)[/tex] in the correct quotient should be [tex]\(24\)[/tex].
Hence, the term containing an error in the quotient is:
[tex]\[
29x
\][/tex]
The correct term should be:
[tex]\[
24x
\][/tex]
So, the term [tex]\(29x\)[/tex] in the given quotient contains an error. Here's the final answer in context:
[tex]\[
3x^3 - 6x^2 + 29x - 91 + \frac{356}{x + 4}
\][/tex]
The correct term is:
[tex]\[
24x
\][/tex]
Therefore, the term containing the error in the given quotient is:
[tex]\[
29x
\][/tex]
