Use the binomial theorem to expand your binomial expression, and state the range of values of x for which the expansion is valid. (1 + 𝑥)−2

To expand the binomial expression (1+x)^{-2} using the binomial theorem, we have the formula:
(1 + x)^{-n} = \sum_{r=0}^{\infty} \binom{-n}{r}x^r = 1 - nx + \frac{n(n+1)}{2!}x^2 - \frac{n(n+1)(n+2)}{3!}x^3 + \ldots
Here, n = 2, so
(1 + x)^{-2} = 1 - 2x + \frac{2 \cdot 3}{2}x^2 - \frac{2 \cdot 3 \cdot 4}{3 \cdot 2}x^3 + \ldots
(1 + x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots

The expansion is valid for all values of x such that the absolute value of x is less than 1, or |x| < 1.

\boxed{(1 + x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots, \text{ for } |x| < 1}