I think I can give it a go!
Proof : By the division theorem any number can be expressed in one of the forms 4q,4q+1,4q+2,4q+3
After squaring each of the odd expressions :
\((4q+1)^2= 16q^2+8q+1=8(2q^2+q)+1 \)
As shown here, it can be expressed as 8 times a number, then added by one.
I hope this helps!