1. **Problem:** Factor the trinomial $$45x^2 - 255xy + 350y^2$$ completely. 2. **Step 1: Identify the coefficients.** - $$a = 45$$ (coefficient of $$x^2$$) - $$b = -255$$ (coefficient of $$xy$$) - $$c = 350$$ (coefficient of $$y^2$$) 3. **Step 2: Find the greatest common factor (GCF).** - GCF of 45, 255, and 350 is 5. 4. **Step 3: Factor out the GCF.** $$45x^2 - 255xy + 350y^2 = 5(9x^2 - 51xy + 70y^2)$$ 5. **Step 4: Factor the trinomial inside the parentheses.** - Multiply $$a$$ and $$c$$: $$9 \times 70 = 630$$ - Find two numbers that multiply to 630 and add to $$b = -51$$. - These numbers are $$-30$$ and $$-21$$ because $$-30 \times -21 = 630$$ and $$-30 + -21 = -51$$. 6. **Step 5: Rewrite the middle term using these numbers.** $$9x^2 - 30xy - 21xy + 70y^2$$ 7. **Step 6: Factor by grouping.** $$= (9x^2 - 30xy) + (-21xy + 70y^2)$$ $$= 3x(3x - 10y) - 7y(3x - 10y)$$ 8. **Step 7: Factor out the common binomial.** $$= (3x - 7y)(3x - 10y)$$ 9. **Step 8: Write the complete factorization including the GCF.** $$45x^2 - 255xy + 350y^2 = 5(3x - 7y)(3x - 10y)$$ **Final answer:** $$\boxed{5(3x - 7y)(3x - 10y)}$$