To match the rational expressions with their rewritten forms, we need to analyze each expression and rewrite it in a simplified fractional form. Let's do this step-by-step for each given expression:
1. Expression: [tex]\(\frac{x^2 + x + 4}{x-2}\)[/tex]
- Rewrite [tex]\(x^2 + x + 4\)[/tex] as follows:
[tex]\[
x^2 + x + 4 = (x-2)(x+3) + 10
\][/tex]
- Thus, we can rewrite:
[tex]\[
\frac{x^2 + x + 4}{x-2} = x + 3 + \frac{10}{x-2}
\][/tex]
2. Expression: [tex]\(\frac{x^2 - x + 4}{x-2}\)[/tex]
- Rewrite [tex]\(x^2 - x + 4\)[/tex] as follows:
[tex]\[
x^2 - x + 4 = (x-2)(x+1) + 6
\][/tex]
- Thus, we can rewrite:
[tex]\[
\frac{x^2 - x + 4}{x-2} = x + 1 + \frac{6}{x-2}
\][/tex]
3. Expression: [tex]\(\frac{x^2 - 4x + 10}{x-2}\)[/tex]
- Rewrite [tex]\(x^2 - 4x + 10\)[/tex] as follows:
[tex]\[
x^2 - 4x + 10 = (x-2)(x-2) + 6
\][/tex]
- Thus, we can rewrite:
[tex]\[
\frac{x^2 - 4x + 10}{x-2} = x - 2 + \frac{6}{x-2}
\][/tex]
4. Expression: [tex]\(\frac{x^2 - 5x + 16}{x-2}\)[/tex]
- Rewrite [tex]\(x^2 - 5x + 16\)[/tex] as follows:
[tex]\[
x^2 - 5x + 16 = (x-2)(x-3) + 10
\][/tex]
- Thus, we can rewrite:
[tex]\[
\frac{x^2 - 5x + 16}{x-2} = x - 3 + \frac{10}{x-2}
\][/tex]
Based on these rewrites, we can match the expressions to their rewritten forms as follows:
[tex]\[
\begin{array}{l}
(x-2)+\frac{6}{x-2} \longrightarrow \frac{x^2 - 4x + 10}{x-2} \quad \text{(Expression 3)} \\
(x+3)+\frac{10}{x-2} \longrightarrow \frac{x^2 + x + 4}{x-2} \quad \text{(Expression 1)} \\
(x+1)+\frac{6}{x-2} \longrightarrow \frac{x^2 - x + 4}{x-2} \quad \text{(Expression 2)} \\
(x-3)+\frac{10}{x-2} \longrightarrow \frac{x^2 - 5x + 16}{x-2} \quad \text{(Expression 4)} \\
\end{array}
\][/tex]
