1. **State the problem:** Solve the quadratic equation $3x^2 + 4 = 0$ for $x$. 2. **Formula and rules:** The general form of a quadratic equation is $ax^2 + bx + c = 0$. Here, $a=3$, $b=0$, and $c=4$. 3. **Isolate the quadratic term:** $$3x^2 + 4 = 0$$ $$3x^2 = -4$$ 4. **Divide both sides by 3:** $$\cancel{3}x^2 = \frac{-4}{\cancel{3}}$$ $$x^2 = -\frac{4}{3}$$ 5. **Take the square root of both sides:** $$x = \pm \sqrt{-\frac{4}{3}}$$ 6. **Simplify the square root:** $$x = \pm \sqrt{-1 \times \frac{4}{3}} = \pm \sqrt{-1} \times \sqrt{\frac{4}{3}} = \pm i \times \frac{2}{\sqrt{3}}$$ 7. **Rationalize the denominator:** $$x = \pm i \times \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \pm \frac{2\sqrt{3}}{3}i$$ **Final answer:** $$x = \pm \frac{2\sqrt{3}}{3}i$$