To simplify the fraction
\[ \frac{1 + 4x^{-1} - 21x^{-2}}{1 - 2x^{-1} - 63x^{-2}}, \]
we start by rewriting the expression in the numerator and the denominator in terms of positive powers of \( x \).
First, we can express \( x^{-1} \) and \( x^{-2} \) as \( \frac{1}{x} \) and \( \frac{1}{x^2} \) respectively.
- Rewriting the numerator:
\[ 1 + 4x^{-1} - 21x^{-2} = 1 + \frac{4}{x} - \frac{21}{x^2} = \frac{x^2 + 4x - 21}{x^2}. \]
- Rewriting the denominator:
\[ 1 - 2x^{-1} - 63x^{-2} = 1 - \frac{2}{x} - \frac{63}{x^2} = \frac{x^2 - 2x - 63}{x^2}. \]
Now we can substitute these simplifications back into our fraction:
\[ \frac{\frac{x^2 + 4x - 21}{x^2}}{\frac{x^2 - 2x - 63}{x^2}}. \]
When dividing by a fraction, we can multiply by its reciprocal:
\[ \frac{x^2 + 4x - 21}{x^2 - 2x - 63}. \]
Next, we simplify \( x^2 + 4x - 21 \) and \( x^2 - 2x - 63 \) by factoring.
- Factoring the numerator:
We can look to factor \( x^2 + 4x - 21 \): \[ x^2 + 4x - 21 = (x + 7)(x - 3). \]
- Factoring the denominator:
Next, we factor \( x^2 - 2x - 63 \): \[ x^2 - 2x - 63 = (x - 9)(x + 7). \]
Thus, our simplified fraction now looks like this:
\[ \frac{(x + 7)(x - 3)}{(x - 9)(x + 7)}. \]
Now, we can cancel the common factor \( (x + 7) \):
\[ \frac{x - 3}{x - 9}, \]
provided that \( x \neq -7 \) (which avoids division by zero).
Finally, the simplified form of the original expression is
\[ \frac{x - 3}{x - 9}, \]
where \( x \neq 9 \) and \( x \neq -7 \).