To subtract the given rational expressions, we need to ensure they have a common denominator. The expressions you provided are:
\[ \frac{x^2 - 2}{4x - 7} - \frac{x + 4}{4x - 7} \]
Since both fractions already have the same denominator \(4x - 7\), we can directly subtract the numerators:
\[ \frac{(x^2 - 2) - (x + 4)}{4x - 7} \]
Now, simplify the numerator:
\[ x^2 - 2 - x - 4 = x^2 - x - 6 \]
So, we can write the expression as:
\[ \frac{x^2 - x - 6}{4x - 7} \]
Next, we can try to factor the numerator \(x^2 - x - 6\):
To factor \(x^2 - x - 6\), we look for two numbers that multiply to \(-6\) and add to \(-1\). The numbers \(-3\) and \(2\) fit this.
Thus, we factor it as:
\[ x^2 - x - 6 = (x - 3)(x + 2) \]
Now, substituting back, we obtain:
\[ \frac{(x - 3)(x + 2)}{4x - 7} \]
The final result of the subtraction is:
\[ \frac{(x - 3)(x + 2)}{4x - 7} \]
This is the difference of the two rational expressions.
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