1. **State the problem:** Solve for $x$ in the equation $$\frac{1 - x^2}{1 - x} = \frac{2}{1}.$$\n\n2. **Recall the formula and rules:** We want to solve the equation by simplifying the left side and then equating it to 2. Note that $1 - x^2$ is a difference of squares and can be factored as $$(1 - x)(1 + x).$$\n\n3. **Simplify the left side:** Substitute the factorization into the fraction:\n$$\frac{1 - x^2}{1 - x} = \frac{(1 - x)(1 + x)}{1 - x}.$$\n\n4. **Cancel common factors:** Since $1 - x$ appears in numerator and denominator, we can cancel it:\n$$\frac{\cancel{(1 - x)}(1 + x)}{\cancel{1 - x}} = 1 + x.$$\n\n5. **Set the simplified expression equal to the right side:**\n$$1 + x = 2.$$\n\n6. **Solve for $x$:** Subtract 1 from both sides:\n$$x = 2 - 1 = 1.$$\n\n7. **Check for restrictions:** The original denominator $1 - x$ cannot be zero, so $x \neq 1$. Since our solution $x=1$ makes the denominator zero, it is not valid.\n\n8. **Conclusion:** There is no valid solution for $x$ because the only candidate solution is excluded by the domain restriction.