1. **State the problem:** Solve the quadratic equation $$60x^2 + 25x - 15 = 0$$ by factoring. 2. **Formula and rules:** To solve by factoring, we first factor the quadratic expression into the product of two binomials and then set each factor equal to zero. 3. **Step 1: Factor out the greatest common factor (GCF):** $$60x^2 + 25x - 15 = 5(12x^2 + 5x - 3)$$ 4. **Step 2: Factor the quadratic inside the parentheses:** We look for two numbers that multiply to $$12 \times (-3) = -36$$ and add to $$5$$. These numbers are $$9$$ and $$-4$$ because $$9 \times (-4) = -36$$ and $$9 + (-4) = 5$$. 5. **Step 3: Rewrite the middle term using these numbers:** $$12x^2 + 9x - 4x - 3$$ 6. **Step 4: Factor by grouping:** $$3x(4x + 3) - 1(4x + 3) = (3x - 1)(4x + 3)$$ 7. **Step 5: Write the full factorization:** $$5(3x - 1)(4x + 3) = 0$$ 8. **Step 6: Set each factor equal to zero and solve for $$x$$:** $$3x - 1 = 0 \Rightarrow x = \frac{1}{3}$$ $$4x + 3 = 0 \Rightarrow x = -\frac{3}{4}$$ **Final answer:** $$x = \frac{1}{3}, -\frac{3}{4}$$