$$4a^{2}+4+\frac{1}{a^{2}}$$
$\left(2a+\frac{1}{a}\right)^{2}$
$$\frac{\left(4a^{2}+4\right)a^{2}}{a^{2}}+\frac{1}{a^{2}}$$
$$\frac{\left(4a^{2}+4\right)a^{2}+1}{a^{2}}$$
$$\frac{4a^{4}+4a^{2}+1}{a^{2}}$$
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$\frac{\left(2a^{2}+1\right)^{2}}{a^{2}}$
