To find the difference of the rational expressions
\[ \frac{x^2 - 2}{x^2 - 5} - \frac{4x + 1}{5 - x^2} + \frac{?}{x^2 - 5} \]
we first notice that \(5 - x^2\) can be rewritten as \(- (x^2 - 5)\). Thus, we rewrite the second term:
\[ \frac{4x + 1}{5 - x^2} = \frac{4x + 1}{-(x^2 - 5)} = -\frac{4x + 1}{x^2 - 5} \]
Now we can combine the first two expressions:
\[ \frac{x^2 - 2}{x^2 - 5} - \left(-\frac{4x + 1}{x^2 - 5}\right) = \frac{x^2 - 2 + 4x + 1}{x^2 - 5} \]
Combine the numerators:
\[ x^2 + 4x + 1 - 2 = x^2 + 4x - 1 \]
So we have:
\[ \frac{x^2 + 4x - 1}{x^2 - 5} \]
Now, we can include the third term (which we need to find) into our expression. Let \(y\) be the numerator of the third term:
\[ \frac{x^2 + 4x - 1}{x^2 - 5} + \frac{y}{x^2 - 5} = \frac{x^2 + 4x - 1 + y}{x^2 - 5} \]
To find \(y\), we want the entire expression to be in the form:
\[ \frac{x^2 + 4x - 1 + y}{x^2 - 5} \]
If we compare this to the final expression, it seems like \(y\) should be whatever we want to be added to \(x^2 + 4x - 1\) to achieve a certain result. There isn't an explicit target for \(y\) given in the problem statement; therefore, without constraints, \(y\) could be chosen based on what's intended.
For instance, if we define that the expression should equal to zero, we can set:
\[ x^2 + 4x - 1 + y = 0 \Rightarrow y = - (x^2 + 4x - 1) = -x^2 - 4x + 1 \]
Thus the simplified expression becomes:
\[ \frac{0}{x^2 - 5} = 0 \]
If no specific condition for \(y\) is given, you are free to choose \(y\) based on your needs. Please provide additional information if there is a specific condition for the result.
Therefore,
\[ \text{Final expression: } \frac{x^2 + 4x - 1 + y}{x^2 - 5} \]
Where \(y\) can be defined as per your requirements.
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