The given polygon is a square which means its all sides will be equal.
According to the above statement:-
[tex] \tt \: x ^{2} + x = x ^{2} + 4[/tex]
Let's solve it ~
[tex] \tt \cancel{x ^{2} } + x = \cancel{x ^{2} } + 4[/tex]
[tex] \tt \: x = 4[/tex]
[tex]\rule{300pt}{3pt}[/tex]
[tex]\huge{|\underline{\mathtt{\red{S}\blue{i}\orange{d}\pink{e}\blue{s}\purple{:}\green{-}\red{}\blue{}\orange{}}}}[/tex]
[tex] \sf \: x ^{2} + x \\ \sf \: 4 ^{2} + 4 \\ \sf \: 16 + 4 \\ \sf \: 20[/tex]
Thus, The sides of the polygon will be 20cm each!
