Completely factor the polynomial, if possible.
\[
4 x^{2}+20 x y+25 y^{2}
\]

Answer
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Step 1 ：The given polynomial is a quadratic expression in two variables x and y. To factorize it, we need to identify if it is a perfect square trinomial. A perfect square trinomial is a trinomial that is the square of a binomial. The general form of a perfect square trinomial is \(a^{2} + 2ab + b^{2}\), which can be factorized to \((a+b)^{2}\).

Step 2 ：Looking at the given polynomial, we can see that it is in the form of \(a^{2} + 2ab + b^{2}\), where \(a = 2x\), \(b = 5y\). Therefore, we can factorize it as \((2x + 5y)^{2}\).

Step 3 ：Final Answer: The factored form of the polynomial \(4 x^{2}+20 x y+25 y^{2}\) is \(\boxed{(2x + 5y)^{2}}\).