4. Given that the line $l$ is the tangent line to the graph of the function $f(x)=2 \ln x+x^{2}$, when the slope of $l$ is the smallest, the equation of $l$ is
$f^{\prime}(x)=\frac{2}{x}+2 x \geqslant 4$, equality holds when $x=1$. Therefore, the equation of the tangent line is $y=4 x-3$.