1. **State the problem:** Find the range of values of the constant $k$ for which the line $y = kx$ intersects the curve $y = x^2 + 3kx + 2 - k$ at two distinct points. 2. **Set up the equation for intersection:** At intersection points, the $y$ values are equal, so set the line equal to the curve: $$kx = x^2 + 3kx + 2 - k$$ 3. **Rearrange the equation:** Move all terms to one side: $$0 = x^2 + 3kx + 2 - k - kx$$ $$0 = x^2 + (3k - k)x + 2 - k$$ $$0 = x^2 + 2kx + 2 - k$$ 4. **Analyze the quadratic equation:** The quadratic in $x$ is $$x^2 + 2kx + (2 - k) = 0$$ For two distinct intersection points, this quadratic must have two distinct real roots. 5. **Use the discriminant condition:** The discriminant $\Delta$ is $$\Delta = (2k)^2 - 4 \times 1 \times (2 - k) = 4k^2 - 4(2 - k) = 4k^2 - 8 + 4k$$ 6. **Simplify the discriminant:** $$\Delta = 4k^2 + 4k - 8$$ 7. **Condition for two distinct real roots:** $$\Delta > 0$$ $$4k^2 + 4k - 8 > 0$$ 8. **Divide both sides by 4:** $$\cancel{4}k^2 + \cancel{4}k - \cancel{8} > 0$$ $$k^2 + k - 2 > 0$$ 9. **Factor the quadratic:** $$k^2 + k - 2 = (k + 2)(k - 1)$$ 10. **Solve inequality:** $$(k + 2)(k - 1) > 0$$ This product is positive when both factors are positive or both are negative. - Both positive: $k - 1 > 0 \Rightarrow k > 1$ - Both negative: $k + 2 < 0 \Rightarrow k < -2$ 11. **Final answer:** $$k < -2 \quad \text{or} \quad k > 1$$ Thus, the line $y = kx$ intersects the curve $y = x^2 + 3kx + 2 - k$ at two distinct points if and only if $k < -2$ or $k > 1$.