To add the rational expressions \(\frac{x^2 - 23x - 2}{3x - 2} + \frac{x + 5}{3x - 2}\), we first note that they have a common denominator of \(3x - 2\). Therefore, we can combine the numerators directly.
This gives us:
\[ \frac{(x^2 - 23x - 2) + (x + 5)}{3x - 2} \]
Now, we simplify the numerator:
- Combine like terms:
- The \(x^2\) term: \(x^2\)
- The \(x\) terms: \(-23x + x = -22x\)
- The constant terms: \(-2 + 5 = 3\)
So the combined numerator is:
\[ x^2 - 22x + 3 \]
Thus, the expression becomes:
\[ \frac{x^2 - 22x + 3}{3x - 2} \]
Now, let's look at the provided response options:
- Option 1: \(\frac{x^3 + 3}{3x - 2}\)
- Option 2: \(\frac{x^2 + x + 3}{6x - 4}\)
- Option 3: \(\frac{x^2 + x + 7}{3x - 2}\)
- Option 4: \(\frac{x^2 + x + 3}{3x - 2}\)
From our simplification, the numerator \(x^2 - 22x + 3\) does not match any of the options listed. Thus, let’s verify the correct series of combining terms to see if a mistake was made.
Returning to our simplified expression, let's check:
\[ x^2 - 22x + 3 \]
This strongly appears to be incorrect based on the choices provided, leading us to suspect that the problem might expect further manipulations or matches with those forms.
However, from the analysis, without combining incorrectly:
- The \(3x - 2\) denominator remains unchanged.
- The numerator \(x^2 - 22x + 3\) does not correlate neatly into any of the answer forms either based on constant values or coefficients.
Would you like to verify stepwise or simplify further? Let me know, and we can continue examining valid pathways from here if required!
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