To subtract the given rational expressions \( \frac{x^2 + 4x - 21}{x^2 - 2x - 3} - 4x + 1 \), we first need to simplify both parts.
- Factor the expression:
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Numerator: \( x^2 + 4x - 21 \)
- This can be factored as \( (x + 7)(x - 3) \).
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Denominator: \( x^2 - 2x - 3 \)
- This factors to \( (x - 3)(x + 1) \).
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So, we have: \[ \frac{(x + 7)(x - 3)}{(x - 3)(x + 1)} - (4x + 1) \]
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Simplifying the expression:
- The fraction can be simplified partially: \[ \frac{x + 7}{x + 1} - (4x + 1) \]
- The next step is to have a common denominator. The common denominator between \( \frac{x + 7}{x + 1} \) and \( 4x + 1 \) is: \[ (x + 1) \]
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Rewrite the second term over the common denominator:
- \( 4x + 1 \) can be expressed as: \[ \frac{(4x + 1)(x + 1)}{(x + 1)} = \frac{4x^2 + 4x + x + 1}{(x + 1)} = \frac{4x^2 + 5x + 1}{(x + 1)} \]
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Now combine the two fractions:
- We have: \[ \frac{(x + 7) - (4x^2 + 5x + 1)}{(x + 1)} \]
- Simplifying the numerator: \[ x + 7 - 4x^2 - 5x - 1 = -4x^2 - 4x + 6 \]
- Which can be factored to: \[ -4(x^2 + x - 1) \]
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Putting it all together, we have: \[ \frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)} \]
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Setting this into the form needed for answer choices, we see:
- Perform further simplification of the numerator if possible or compare to choices given.
To compare this expression with the choices given:
- None of the exact forms seem to match; however, we should make sure that we have:
- If simplified correctly it should yield one of the expected forms.
Thus, considering the achievable options, if we proceed to validate specific forms: We should check if: \[ x^2 - 25 = 0 \implies -25 \text{ implies no reach for the standard } \] Hence providing no similar match remains intact.
Finally, consider if the original subtraction matches one choice:
After further factoring or organizing if none meet but more evaluative checks are needed.
The answer might imply a direct rational reduction and hence choices should be reviewed eminently. Upon full evaluation direct crossover equivalent must be validated among options routes.
However: \[ \text{Final numerical assessment shows } => \frac{-4(x^2 + x - 1)}{(x + 1)(x - 3)} \]
Since no particular match was met but approximates nearby yet not overlap choices manifest. For final ascertain check original must revert through until clean.