1. **State the problem:** Solve the quadratic equation $3x^2 + 2x + 3 = 5$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$3x^2 + 2x + 3 - 5 = 0$$ which simplifies to $$3x^2 + 2x - 2 = 0$$ 3. **Identify coefficients:** For the quadratic equation $ax^2 + bx + c = 0$, here $a=3$, $b=2$, and $c=-2$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = 2^2 - 4 \times 3 \times (-2) = 4 + 24 = 28$$ 6. **Substitute values into the formula:** $$x = \frac{-2 \pm \sqrt{28}}{2 \times 3} = \frac{-2 \pm \sqrt{28}}{6}$$ 7. **Simplify the square root:** $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ 8. **Rewrite the expression:** $$x = \frac{-2 \pm 2\sqrt{7}}{6}$$ 9. **Cancel common factor 2 in numerator and denominator:** $$x = \frac{\cancel{2}(-1 \pm \sqrt{7})}{\cancel{2} \times 3} = \frac{-1 \pm \sqrt{7}}{3}$$ 10. **Final answer:** $$x = \frac{-1 + \sqrt{7}}{3} \quad \text{or} \quad x = \frac{-1 - \sqrt{7}}{3}$$