$\rightarrow \left\{ \begin{array}{l} a\geq1-b \\b\geq 1-a\end{array} \right.$
$\rightarrow P=2a+\frac{b}{4a}+b^2\geq 2a+\frac{1}{4a}-\frac{1}{4}+b^2\geq (a+\frac{1}{4a})+a+b^2-\frac{1}{4}$
$\geq(a+\frac{1}{4a})+(b^2-b+\frac{1}{4})+\frac{1}{2}$
$\geq \frac{3}{2} $