1. **State the problem:** Factor the quadratic expression $$15x^2 + 14x + 3$$ by grouping using the ac-method. 2. **Formula and rules:** The ac-method involves multiplying the coefficient of $$x^2$$ (which is 15) by the constant term (which is 3). Then, find two numbers that multiply to $$15 \times 3 = 45$$ and add to the middle coefficient, 14. 3. **Find the two numbers:** The pair of numbers that multiply to 45 and add to 14 are 9 and 5. 4. **Rewrite the middle term:** Split the middle term $$14x$$ into $$9x + 5x$$. 5. **Rewrite the expression:** $$15x^2 + 9x + 5x + 3$$ 6. **Group terms:** $$(15x^2 + 9x) + (5x + 3)$$ 7. **Factor each group:** $$3x(5x + 3) + 1(5x + 3)$$ 8. **Factor out the common binomial:** $$(3x + 1)(5x + 3)$$ 9. **Choose the correct form:** Since the middle term was split into $$+9x + 5x$$, the form is: $$15x^2 + [9] x + [5] x + 3$$ 10. **Final factorization:** $$\boxed{(3x + 1)(5x + 3)}$$