(Originais Teachy 2023) - Question Very Hard of Mathematics

An architect is designing a new square that will have a large decorative panel. The panel will be composed of square tiles with dimensions of 1 meter by 1 meter. To create a symmetrical pattern, the architect wants to use the algebraic expression (x+1)^2 - x^2 to determine the number of tiles that will form one side of the panel, with x being the number of tiles already installed on one side. Knowing that the quantity of tiles already installed is 100 and that the architect wants the side of the panel to have a whole number of tiles, calculate how many tiles the architect should install on one side of the panel for the expression to be a perfect square. Explain the concept of the notable product used in the simplification of the algebraic expression to determine the number of tiles needed, and discuss how this concept relates to the idea of a perfect square applied to the panel project.