1. **State the problem:** Simplify or analyze the expression $kx^2 - 2x - k$. 2. **Identify the expression:** This is a quadratic expression in terms of $x$ with coefficients involving $k$. 3. **Rewrite the expression:** $$kx^2 - 2x - k$$ 4. **Factor the expression if possible:** We look for factors of the form $(ax + b)(cx + d)$ such that: $$ac = k, \quad bd = -k, \quad ad + bc = -2$$ 5. **Try factoring:** Assuming $a = k$ and $c = 1$ (or vice versa), test: $$(kx + m)(x + n) = kx^2 + (kn + m)x + mn$$ We want: $$kn + m = -2$$ $$mn = -k$$ 6. **Solve for $m$ and $n$:** From $mn = -k$, let $m = -\frac{k}{n}$. Substitute into $kn + m = -2$: $$kn - \frac{k}{n} = -2$$ Multiply both sides by $n$: $$k n^2 - k = -2 n$$ Rearranged: $$k n^2 + 2 n - k = 0$$ 7. **Solve quadratic in $n$:** $$n = \frac{-2 \pm \sqrt{4 + 4k^2}}{2k} = \frac{-2 \pm 2\sqrt{1 + k^2}}{2k} = \frac{-1 \pm \sqrt{1 + k^2}}{k}$$ 8. **Conclusion:** The factorization depends on $k$ and the roots for $n$ are: $$n = \frac{-1 \pm \sqrt{1 + k^2}}{k}$$ Thus, the factorization is: $$(kx + m)(x + n)$$ with $m = -\frac{k}{n}$ and $n$ as above. This completes the factorization process for the quadratic expression.