To subtract the rational expressions \(\frac{x^2 + 4x - 21}{x^2 - 2x - 3} - 4x + 1\), we first need to rewrite \(4x + 1\) as a rational expression that can be combined with the first term.
Step 1: Factor the denominator
First, let's factor the denominator of the first term: \(x^2 - 2x - 3\).
This quadratic can be factored as: \[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
Step 2: Rewrite \(4x + 1\) as a rational expression
We can write \(4x + 1\) over the same denominator: \[ 4x + 1 = \frac{(4x + 1)((x - 3)(x + 1))}{(x - 3)(x + 1)} = \frac{(4x + 1)(x - 3)(x + 1)}{(x - 3)(x + 1)} \] To combine them, we need to express \(4x + 1\) in terms of the common denominator.
Step 3: Find a common denominator and combine
Using the common denominator of \((x-3)(x+1)\), we have: \[ \frac{x^2 + 4x - 21}{(x-3)(x+1)} - \frac{(4x+1)}{1} \]
Multiply \(4x + 1\) by \(\frac{(x-3)(x+1)}{(x-3)(x+1)}\) to combine: \[ \frac{x^2 + 4x - 21 - (4x + 1)(x - 3)(x + 1)}{(x - 3)(x + 1)} \]
Step 4: Expand and simplify
Now we need to simplify \((4x + 1)(x^2 - 2)\): \[ (4x + 1)(x^2 - 2) \] Using the multiplication directly, we get, \[ 4x(x^2 - 2) + 1(x^2 - 2) = 4x^3 - 8x + x^2 - 2 = 4x^3 + x^2 - 8x - 2 \] We can now write, \[ \frac{x^2 + 4x - 21 - (4x^3 + x^2 - 8x - 2)}{(x - 3)(x + 1)} \]
Combine like terms in the numerator: \[ = \frac{-4x^3 + (x^2 - x^2) + (4x + 8x) - 21 + 2}{(x - 3)(x + 1)} = \frac{-4x^3 + 12x - 19}{(x - 3)(x + 1)} \]
Conclusion
The simplified result after subtracting the two rational expressions is: \[ \frac{-4x^3 + 12x - 19}{(x - 3)(x + 1)} \]
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