Let's match each given expression with its quotient and remainder when divided by its corresponding divisor. We'll analyze each pair of expressions and divisors step-by-step.
1. For the first expression:
[tex]\[
\frac{x^2 - x - 30}{x - 6}
\][/tex]
The quotient and remainder for this division are:
[tex]\[
x + 5 \quad (R = 0)
\][/tex]
So, the result for this division is \(x + 5\) with a remainder of 0.
2. For the second expression:
[tex]\[
\frac{x^3 - 2x^2 - 7x - 4}{x^2 + 2x + 1}
\][/tex]
The quotient and remainder for this division are:
[tex]\[
x - 4 \quad (R = 0)
\][/tex]
So, the result for this division is \(x - 4\) with a remainder of 0.
3. For the third expression:
[tex]\[
\frac{x^3 + 2x^2 - 1}{x^2 - x + 1}
\][/tex]
The quotient and remainder for this division are:
[tex]\[
x + 3 \quad (R = 2x - 4)
\][/tex]
So, the result for this division is \(x + 3\) with a remainder of \(2x - 4\).
Now, let's match the expressions with their corresponding quotients and remainders:
- \(\frac{x^2 - x - 30}{x - 6}\) matches with \(x + 5 \text{ with remainder } 0\).
- \(\frac{x^3 - 2x^2 - 7x - 4}{x^2 + 2x + 1}\) matches with \(x - 4 \text{ with remainder } 0\).
- \(\frac{x^3 + 2x^2 - 1}{x^2 - x + 1}\) matches with \(x + 3 \text{ with remainder } 2x - 4\).
Therefore, the correct matches are:
[tex]\[
\begin{array}{l}
\left(x^2-x-30\right) \div(x-6) \quad \rightarrow \quad x+5 \\
\left(x^3-2 x^2-7 x-4\right) \div\left(x^2+2 x+1\right) \quad \rightarrow \quad x-4 \\
\left(x^3+2 x^2-1\right) \div\left(x^2-x+1\right) \quad \rightarrow \quad x+3 \text{ with remainder } 2x-4 \\
\end{array}
\][/tex]
