1. **State the problem:** Factor the quadratic expression $$12q^2 + 17q + 6$$. 2. **Multiply the coefficient of** $$q^2$$ *and* the constant: $$12 \times 6 = 72$$. 3. **Find two numbers whose product is** $$72$$ *and whose sum is the middle coefficient,* $$17$$. These numbers are $$9$$ and $$8$$. 4. **Rewrite the middle term using these numbers:** $$12q^2 + 9q + 8q + 6$$ 5. **Factor by grouping:** $$= 3q(4q + 3) + 2(4q + 3)$$ 6. **Factor out the common binomial:** $$= (3q + 2)(4q + 3)$$ **Final answer:** $$12q^2 + 17q + 6 = (3q + 2)(4q + 3)$$