jmc

Notice that the quantity $4a^2+1$ appears in various forms throughout the expression on the left-hand side. So let $4a^2+1=x$ to simplify the expression to $\frac{7\sqrt{x}-x}{\sqrt{x}+3}$. This still looks messy, so let $\sqrt{x}=y$. Our equation becomes $$\begin{array}{rl}\frac{7y-y^2}{y+3}& =2.\end{array}$$ Clearing denominators, rearranging, and factoring, we find $$\begin{array}{rl}7y-y^2& =2(y+3)\Rightarrow \\ 7y-y^2& =2y+6\Rightarrow \\ 0& =y^2-5y+6\Rightarrow \\ 0& =(y-2)(y-3).\end{array}$$ Thus $y=2$ or $y=3$, so $\sqrt{x}=2,3$ and $x=4$ or $x=9$. Re-substituting, we have $4a^2+1=4$, meaning $4a^2=3$, $a^2=\frac{3}{4}$, and $a=\pm \frac{\sqrt{3}}{2}$. On the other hand we could have $4a^2+1=9$, giving $4a^2=8$, $a^2=2$, and $a=\pm \sqrt{2}$. The greatest possible value of $a$ is $\boxed{\sqrt{2}}$.