Um, look, any (positive) odd prime with $(5|p) = (p | 5) = 1$ can be used to produce a binary quadratic form $f(x,y) = a x^2 + b x y + c y^2,$ or $\langle a,b,c \rangle,$ with particular coefficients $\langle p,\beta,\gamma \rangle$ with discriminant $\Delta = \beta^2 - 4 p \gamma = 5.$ This can then be reduced to $\langle 1,1,-1 \rangle$ or the equivalent $\langle -1,1,1 \rangle,$ showing that we can write $p = u^2 + u v - v^2.$ A small change shows $p = x^2 + 5 x y + 5 y^2,$ maybe you can do that part.
See Numbers representable as $x^2 + 2y^2$
Note that reduction for indefinite forms is a little different from positive. We can arrange $\langle a,b,c \rangle,$ such that $0 < b < \sqrt \Delta$ and
$ \sqrt \Delta - b < 2 |a| < \sqrt \Delta + b$ and there are more than one, a finite number of reduced forms in each equivalence class.
Reduction can be arranged by a finite sequence of these steps: take the "Hessian" matrix of the form $\langle a,b,c \rangle,$ as
$$
H = \left( \begin{array}{cc}
2a & b \\
b & 2c
\end{array}
\right).
$$
Now take the matrix $P \in SL_2 \mathbb Z$ given by
$$
P = \left( \begin{array}{cc}
0 & -1 \\
1 & \delta
\end{array}
\right)
$$
and calculate
$$ G = P^T H P. $$
Now, $G$ is the Hessian matrix of an "equivalent" form (think about how to go back from a Hessian matrix to a form). A correct choice of the integer $\delta$ takes the form closer to reduced, after a few such steps the form is reduced, and further steps take the form through a cycle of equivalent forms, back to the first reduced one. The absolute values of the $\delta$'s (once reduced) are the digits for the repeated part of the continued fraction for a certain quadratic irrational, tied up with Pell's equation. The continued fraction with all "digits" equal to $1$ is the Golden Ratio. I'm just sayin'.
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
1 5 5
0 form 1 5 5 delta 0
1 form 5 -5 1 delta -2
2 form 1 1 -1
-1 2
0 -1
To Return
-1 -2
0 -1
0 form 1 1 -1 delta -1
1 form -1 1 1 delta 1
2 form 1 1 -1
minimum was 1rep 1 0 disc 5 dSqrt 2.2360679775 M_Ratio 5
Automorph, written on right of Gram matrix:
-1 -1
-1 -2
Trace: -3 gcd(a21, a22 - a11, a12) : 1
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
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