1. **State the problem:** Factor the quadratic expression $6x^2 + 11x + 3$ using the decomposition method. 2. **Recall the decomposition method:** We look for two numbers that multiply to $a \times c$ and add to $b$, where the quadratic is $ax^2 + bx + c$. 3. For $6x^2 + 11x + 3$, $a=6$, $b=11$, and $c=3$. Calculate $a \times c = 6 \times 3 = 18$. 4. Find two numbers that multiply to 18 and add to 11. These numbers are 9 and 2 because $9 \times 2 = 18$ and $9 + 2 = 11$. 5. Rewrite the middle term $11x$ as $9x + 2x$: $$6x^2 + 9x + 2x + 3$$ 6. Group terms: $$(6x^2 + 9x) + (2x + 3)$$ 7. Factor out the greatest common factor (GCF) from each group: $$3x(2x + 3) + 1(2x + 3)$$ 8. Notice the common binomial factor $(2x + 3)$, factor it out: $$(3x + 1)(2x + 3)$$ **Final answer:** The factorization of $6x^2 + 11x + 3$ is $$\boxed{(3x + 1)(2x + 3)}$$.