1. **State the problem:** Solve the quadratic equation using the quadratic formula: $$x(2x + 2) = 11$$ 2. **Rewrite the equation in standard form:** $$2x^2 + 2x = 11$$ Subtract 11 from both sides: $$2x^2 + 2x - 11 = 0$$ 3. **Identify coefficients:** For the quadratic equation $$ax^2 + bx + c = 0$$, here: $$a = 2, \quad b = 2, \quad c = -11$$ 4. **Quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4(2)(-11) = 4 + 88 = 92$$ 6. **Simplify the square root:** $$\sqrt{92} = \sqrt{4 \times 23} = 2\sqrt{23}$$ 7. **Substitute values into the formula:** $$x = \frac{-2 \pm 2\sqrt{23}}{2 \times 2} = \frac{-2 \pm 2\sqrt{23}}{4}$$ 8. **Simplify the fraction:** Divide numerator and denominator by 2: $$x = \frac{-1 \pm \sqrt{23}}{2}$$ **Final answer:** $$x = \frac{-1 + \sqrt{23}}{2}, \quad \frac{-1 - \sqrt{23}}{2}$$