1. **State the problem:** Solve the quadratic equation $3x^2 + 4x + 1 = 0$ by factorization. 2. **Recall the factorization method:** For a quadratic equation $ax^2 + bx + c = 0$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a=3$, $b=4$, $c=1$. So, $a \times c = 3 \times 1 = 3$ and we need two numbers that multiply to 3 and add to 4. 4. **Find the numbers:** The numbers are 3 and 1 because $3 \times 1 = 3$ and $3 + 1 = 4$. 5. **Rewrite the middle term:** $$3x^2 + 3x + 1x + 1 = 0$$ 6. **Group terms:** $$ (3x^2 + 3x) + (1x + 1) = 0$$ 7. **Factor each group:** $$3x(x + 1) + 1(x + 1) = 0$$ 8. **Factor out the common binomial:** $$(3x + 1)(x + 1) = 0$$ 9. **Set each factor equal to zero:** $$3x + 1 = 0 \quad \text{or} \quad x + 1 = 0$$ 10. **Solve for $x$:** $$3x = -1 \Rightarrow x = -\frac{1}{3}$$ $$x = -1$$ **Final answer:** The solutions are $x = -\frac{1}{3}$ and $x = -1$.