If $n$ is represented by a quadratic form, then $4an$ is a square modulo $|d|$

Question

If $n$ is represented by a quadratic form $ax^2+bxy+cy^2$ then $4an$ is a square modulo $|d|$.

I got this question from textbook but I wasn't able to see the role of the "a". The following statement is known in our class:

Let $n$ and $d$ be given. $n$ is represented by some quadratic form $f$ Of discriminate $d$ if and only if $d$ is a square mod $4|n|$.

Answer 1

If $n = a x^2 + b x y + c y^2$ (with all variables integers) and $d = b^2-4ac$ the discriminant, then $$4 a n = 4 a^2 x^2 + 4 a b x y + 4 a c y^2 = (2 a x + b y)^2 - y^2 d$$ so this is indeed a square plus a multiple of $|d|$.