1. **State the problem:** Factor the quadratic expression $$16x^2 - 40xy + 25y^2$$. 2. **Recognize the form:** This is a quadratic trinomial in terms of $x$ and $y$. It resembles a perfect square trinomial of the form $$a^2 - 2ab + b^2 = (a - b)^2$$. 3. **Identify $a$ and $b$:** - $a^2 = 16x^2$ so $a = 4x$. - $b^2 = 25y^2$ so $b = 5y$. 4. **Check the middle term:** - The middle term is $-40xy$. - The formula middle term is $-2ab = -2 \times 4x \times 5y = -40xy$, which matches exactly. 5. **Write the factorization:** $$16x^2 - 40xy + 25y^2 = (4x - 5y)^2$$. 6. **Explanation:** Since the expression fits the perfect square trinomial pattern, it factors neatly into the square of a binomial. **Final answer:** $$\boxed{(4x - 5y)^2}$$