Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$
When we have $$Q(x, y)=x^2+y^2 \\\ Q'(x, y)=5x^2+5y^2$$ how can we check if they are equivalent??? How can we find such a $T$ ???
Depending on the base field $K$, this might be a difficult problem in general.
However, for quadratic forms of rank $2$, this is easy. You just have to check that the two quadratic forms have same discriminant, and represents a commun value, for example, which is the case here $5$ is represented by both quadratic forms, and the two determinants are equal to $1$ mod squares.
In your particular case, it is better to notice that $5=1^2+2^2$, and write $(1^2+2^2)(x^2+y^2)=(x-2y)^2+(2x+y)^2$, which provides you everything you want, since the linear forms $x+2y$ and $2x+y$ are independent.