1. **Problem:** Find the points of intersection between the curve $$2x^2 - y + 19 = 0$$ and the line $$y + 11x = 4$$. 2. **Step 1: Express one variable from the line equation.** From $$y + 11x = 4$$, we get $$y = 4 - 11x$$. 3. **Step 2: Substitute $$y$$ into the curve equation.** Substitute $$y = 4 - 11x$$ into $$2x^2 - y + 19 = 0$$: $$2x^2 - (4 - 11x) + 19 = 0$$ 4. **Step 3: Simplify the equation.** $$2x^2 - 4 + 11x + 19 = 0$$ $$2x^2 + 11x + 15 = 0$$ 5. **Step 4: Solve the quadratic equation.** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=11$$, $$c=15$$. Calculate the discriminant: $$\Delta = 11^2 - 4 \times 2 \times 15 = 121 - 120 = 1$$ 6. **Step 5: Find the roots.** $$x = \frac{-11 \pm \sqrt{1}}{2 \times 2} = \frac{-11 \pm 1}{4}$$ 7. **Step 6: Calculate each root.** For $$x_1$$: $$x_1 = \frac{-11 + 1}{4} = \frac{-10}{4} = -\frac{5}{2}$$ For $$x_2$$: $$x_2 = \frac{-11 - 1}{4} = \frac{-12}{4} = -3$$ 8. **Step 7: Find corresponding $$y$$ values using $$y = 4 - 11x$$.** For $$x_1 = -\frac{5}{2}$$: $$y_1 = 4 - 11 \times \left(-\frac{5}{2}\right) = 4 + \frac{55}{2} = \frac{8}{2} + \frac{55}{2} = \frac{63}{2}$$ For $$x_2 = -3$$: $$y_2 = 4 - 11 \times (-3) = 4 + 33 = 37$$ 9. **Final answer:** The points of intersection are $$\left(-\frac{5}{2}, \frac{63}{2}\right)$$ and $$(-3, 37)$$.