Solution
= Solution
Assume $|a|_2>1$, equivalently $v_2(a)<0$, and write
$$
a^4+4=a^4\left(1+\frac4{a^4}\right).
$$
For $f(Y)=Y^2-(1+4/a^4)$ at $Y=1$,
$$
v_2(f(1))=v_2(4/a^4)=2-4v_2(a)\geq6>2v_2(f'(1))=2.
$$
The <strong form of Hensel lemma> therefore gives $y\in\mathbb Q_2$ with $y^2=1+4/a^4$. Taking $X=a^2y$ yields $X^2=a^4+4$. Hence every $a$ with $|a|_2>1$, and in particular every sufficiently large $a$, works.