a, $x^2 -2(m+1)x+m^2 +3m=0$ (*)
(*) có 2 nghiệm khi $\Delta' > 0$
$\Delta'= (m+1)^2 - m^2 - 3m$
$= m^2 + 2m+ 1 - m^2 - 3m$
$= -m+1 > 0$
$\Leftrightarrow m<1$
b,
$x_1+x_2= 2(m+1)$
$x_1.x_2= m^2+3m$
$A= x_1^2 + x_2^2$
$= (x_1+x_2)^2 - 2x_1.x_2$
$= 4(m+1)^2 - 2(m^2 + 3m)$
$= 4m^2+ 8m+ 4- 2m^2- 6m$
$= 2m^2 + 2m+ 4$
$= (\sqrt{2}m)^2 + 2.\sqrt{2}m.\frac{1}{\sqrt{2}} + \frac{1}{2} + \frac{7}{2}$
$= (\sqrt{2}m + \frac{1}{\sqrt{2}})^2 + \frac{7}{2} \ge \frac{7}{2}$
$\Rightarrow min = \frac{7}{2}$