The equation \(3x^2 + 2x + 1 = 0\) is a quadratic equation. To solve it, we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, the coefficients are:
- \(a = 3\)
- \(b = 2\)
- \(c = 1\)

Let's substitute these values into the quadratic formula.

\[
x = \frac{-2 \pm \sqrt{2^2 - 4(3)(1)}}{2(3)}
\]

\[
x = \frac{-2 \pm \sqrt{4 - 12}}{6}
\]

\[
x = \frac{-2 \pm \sqrt{-8}}{6}
\]

Since the discriminant (\(\sqrt{-8}\)) is negative, the solutions will be complex numbers.

\[
x = \frac{-2 \pm \sqrt{-8}}{6} = \frac{-2 \pm 2i\sqrt{2}}{6} = \frac{-1 \pm i\sqrt{2}}{3}
\]

Thus, the solutions are:

\[
x = \frac{-1 + i\sqrt{2}}{3} \quad \text{and} \quad x = \frac{-1 - i\sqrt{2}}{3}
\]

These are the complex roots of the quadratic equation.

Would you like a detailed explanation of how we arrived at these roots or have any other questions?

### Related Questions:
1. How do you determine the nature of the roots using the discriminant?
2. What are complex numbers and how are they represented?
3. Can a quadratic equation have real roots if the discriminant is zero?
4. How does the quadratic formula derive from completing the square?
5. What is the geometric interpretation of complex roots?

**Tip:** Always check the discriminant first to determine if the roots will be real or complex.

Learn how to solve the quadratic equation 3x^2 + 2x + 1 = 0 with complex roots using the quadratic formula. This problem involves understanding complex numbers, discriminant analysis, and quadratic equation solutions. Suitable for high school students.

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