If the roots of ${\mathrm{a}}^2{\mathrm{x}}^2+2\mathrm{bx}+{\mathrm{c}}^2=0$ are imaginary then the roots of $\mathrm{b}\left({\mathrm{x}}^2+1\right)+2\mathrm{acx}=0$ are

The given equation is ${\mathrm{a}}^2{\mathrm{x}}^2+2\mathrm{bx}+{\mathrm{c}}^2=0$

If the roots are imaginary, then the discriminant of the quadratic equation is negative.

hence, $b^2-4ac<0$

it implies that $4{\mathrm{b}}^2-4{\mathrm{a}}^2{\mathrm{c}}^2<0\Rightarrow {\mathrm{b}}^2<{\mathrm{a}}^2{\mathrm{c}}^2\Rightarrow {\mathrm{a}}^2{\mathrm{c}}^2-{\mathrm{b}}^2>0$

The second equation is $\mathrm{b}\left({\mathrm{x}}^2+1\right)+2\mathrm{acx}=0$

This can be rewrite as $b{x}^2+2acx+b=0$

its discriminant is $4a^2{c}^2-4b^2=4\left(a^2{c}^2-b^2\right)>0$

Since the discriminat is posiitve, the roots are Real and unequal. 

Therefore, the roots of the equation $b{x}^2+2acx+b=0$ are $\boxed{Real and unequal}$