1. **State the problem:** Solve the equation $$3(2x+1)^2 = 2(5x+1)$$ for $x$. 2. **Use the formula and rules:** Expand the squared term and simplify both sides. 3. **Expand the left side:** $$3(2x+1)^2 = 3(4x^2 + 4x + 1) = 12x^2 + 12x + 3$$ 4. **Rewrite the equation:** $$12x^2 + 12x + 3 = 2(5x + 1)$$ 5. **Expand the right side:** $$2(5x + 1) = 10x + 2$$ 6. **Bring all terms to one side:** $$12x^2 + 12x + 3 - 10x - 2 = 0$$ 7. **Simplify:** $$12x^2 + (12x - 10x) + (3 - 2) = 0$$ $$12x^2 + 2x + 1 = 0$$ 8. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=12$, $b=2$, $c=1$. 9. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 12 \times 1 = 4 - 48 = -44$$ 10. **Interpret the discriminant:** Since $$\Delta < 0$$, there are no real solutions, only complex solutions. 11. **Find the complex solutions:** $$x = \frac{-2 \pm \sqrt{-44}}{24} = \frac{-2 \pm \sqrt{44}i}{24} = \frac{-2 \pm 2\sqrt{11}i}{24}$$ 12. **Simplify the fraction:** $$x = \frac{-2}{24} \pm \frac{2\sqrt{11}i}{24} = -\frac{1}{12} \pm \frac{\sqrt{11}}{12}i$$ **Final answer:** $$x = -\frac{1}{12} \pm \frac{\sqrt{11}}{12}i$$