1. **State the problem:** We need to factor the polynomial $$36x^2y^4 + 12x^3y^2 + 12x^4y^3$$ and match it with the given factors $$-4xy$$ and $$12x^2y^2$$. 2. **Identify the greatest common factor (GCF):** Look for the common factors in each term. - Coefficients: GCF of 36, 12, and 12 is 12. - Variables: For $$x^2, x^3, x^4$$ the minimum power is $$x^2$$. - For $$y^4, y^2, y^3$$ the minimum power is $$y^2$$. So, the GCF is $$12x^2y^2$$. 3. **Factor out the GCF:** $$ 36x^2y^4 + 12x^3y^2 + 12x^4y^3 = 12x^2y^2(\frac{36x^2y^4}{12x^2y^2} + \frac{12x^3y^2}{12x^2y^2} + \frac{12x^4y^3}{12x^2y^2}) $$ 4. **Simplify inside the parentheses:** $$ = 12x^2y^2(3y^2 + x + x^2y) $$ 5. **Check if the polynomial inside parentheses can be factored further:** - The terms are $$3y^2$$, $$x$$, and $$x^2y$$. - No common factors or simple factorization applies here. 6. **Match with given factors:** - The factor $$12x^2y^2$$ matches the GCF factored out. - The other factor is the remaining polynomial $$3y^2 + x + x^2y$$. 7. **Regarding $$-4xy$$:** - It is not a factor of the given polynomial since it does not divide all terms evenly. **Final answer:** $$36x^2y^4 + 12x^3y^2 + 12x^4y^3 = 12x^2y^2(3y^2 + x + x^2y)$$ **Summary:** - Polynomial: $$36x^2y^4 + 12x^3y^2 + 12x^4y^3$$ - Factor: $$12x^2y^2$$ - Remaining factor: $$3y^2 + x + x^2y$$ - $$-4xy$$ is not a factor of the polynomial.