∵P=x²+2xy+2y²+2x+4y+5
=（x²+2xy+y²+2x+2y+1）+（y²+2y+1）+3
=（x+y+1）²+（y+1）²+3≥3．
∴P=x²+2xy+2y²+2x+4y+5的值恒为正数且有最小值3．
故选：A．