4. Represent the numerical expression $2 \cdot 2009^{2}+2 \cdot 2010^{2}$ as the sum of squares of two natural numbers.
Answer: $4019^{2}+1^{2}$ or $2911^{2}+2771^{2}$ Solution. It is sufficient to note that $2 \cdot 2009^{2}+2 \cdot 2010^{2}=(2010+2009)^{2}+(2010-2009)^{2}$.

Note. It can be shown that there are no other answers.

Grading Guidelines. For full credit, it is sufficient to provide and justify one of the two possible answers. A correct answer without justification - 0 points. The correct formula is provided (for example, $(a+b)^{2}+(a-b)^{2}=2 a^{2}+2 b^{2}$ or $\left.2 a^{2}+2(a+1)^{2}=(2 a+1)^{2}+1\right)$, but the answer is incorrect due to a computational error - 5 points.