[ Identical Transformations ]

Represent the numerical expression $2 \cdot 2009^{2}+2 \cdot 2010^{2}$ as the sum of squares of two natural numbers.

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The first method. Let $a=2009.2 \cdot 2009^{2}+2 \cdot 2010^{2}=2 a^{2}+2(a+1)^{2}=4 a^{2}+4 a+2=(2 a+1)^{2}+1^{2}=4019^{2}+1^{2}$. The second method. Using the formula $2 a^{2}+2 b^{2}=(a+b)^{2}+(a-b)^{2}$, we get the same result.

## Answer

$4019^{2}+1^{2}$ or $2911^{2}+2771^{2}$