Factor Polynomial 9Cfe26

1. **State the problem:** Factor the expression $$64 m^{4} + 80 m^{2} n^{4} + 25 n^{8}$$. 2. **Recognize the form:** This is a quadratic in terms of $$m^{2}$$ and $$n^{4}$$. Let $$x = m^{2}$$ and $$y = n^{4}$$, then the expression becomes $$64 x^{2} + 80 x y + 25 y^{2}$$. 3. **Use the quadratic trinomial factoring formula:** For $$a x^{2} + b x y + c y^{2}$$, factor as $$(p x + q y)^2$$ if $$b^{2} = 4 a c$$. 4. **Check discriminant:** $$b^{2} = 80^{2} = 6400$$ and $$4 a c = 4 imes 64 imes 25 = 6400$$, so it is a perfect square. 5. **Find factors:** Since $$a = 64 = 8^{2}$$ and $$c = 25 = 5^{2}$$, the factorization is $$(8 x + 5 y)^2$$. 6. **Substitute back:** Replace $$x = m^{2}$$ and $$y = n^{4}$$ to get $$(8 m^{2} + 5 n^{4})^{2}$$. **Final answer:** $$64 m^{4} + 80 m^{2} n^{4} + 25 n^{8} = (8 m^{2} + 5 n^{4})^{2}$$.