To find the difference of the given rational expressions, we need to simplify the expression step by step.
The given expression is:
\[ \frac{x^2 - 2}{x^2 - 5} - \frac{4x + 15}{-x^2} \]
First, let's simplify the second term \(\frac{4x + 15}{-x^2}\):
\[ \frac{4x + 15}{-x^2} = -\frac{4x + 15}{x^2} \]
Now we can rewrite the expression:
\[ \frac{x^2 - 2}{x^2 - 5} + \frac{4x + 15}{x^2} \]
Next, we need a common denominator to combine these two fractions. The common denominator will be:
\[ (x^2 - 5)x^2 \]
Now, we will rewrite each fraction using the common denominator:
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For the first fraction: \[ \frac{x^2 - 2}{x^2 - 5} = \frac{(x^2 - 2)x^2}{(x^2 - 5)x^2} = \frac{x^4 - 2x^2}{(x^2 - 5)x^2} \]
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For the second fraction: \[ \frac{4x + 15}{x^2} = \frac{(4x + 15)(x^2 - 5)}{(x^2)(x^2 - 5)} = \frac{(4x + 15)(x^2 - 5)}{(x^2 - 5)x^2} \]
Now, we can combine the fractions:
\[ \frac{x^4 - 2x^2 + (4x + 15)(x^2 - 5)}{(x^2 - 5)x^2} \]
Next, we need to expand \((4x + 15)(x^2 - 5)\):
\[ (4x + 15)(x^2 - 5) = 4x \cdot x^2 + 15 \cdot x^2 - 4x \cdot 5 - 15 \cdot 5 = 4x^3 + 15x^2 - 20x - 75 \]
Now, substitute this back into our combined expression:
\[ \frac{x^4 - 2x^2 + 4x^3 + 15x^2 - 20x - 75}{(x^2 - 5)x^2} \]
Combine the like terms in the numerator:
\[ x^4 + 4x^3 + (-2x^2 + 15x^2) - 20x - 75 = x^4 + 4x^3 + 13x^2 - 20x - 75 \]
Thus, the final expression is:
\[ \frac{x^4 + 4x^3 + 13x^2 - 20x - 75}{(x^2 - 5)x^2} \]
This is the simplified result of the difference of the rational expressions.
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