[tex]\mathbf A^2+4\mathbf A-5\mathbf I=0[/tex]
[tex]\implies\mathbf A^2+4\mathbf A+4\mathbf I=(\mathbf A+2\mathbf I)^2=9\mathbf I[/tex]
[tex]\implies\det\bigg((\mathbf A+2\mathbf I)^2\bigg)=\det(9\mathbf I)[/tex]
[tex]\implies\det(\mathbf A+2\mathbf I)^2=9^4\det\mathbf I[/tex]
[tex]\implies\det(\mathbf A+2\mathbf I)=\pm\sqrt{9^4}[/tex]
[tex]\implies\det(\mathbf A+2\mathbf I)=9^2=81[/tex]
where we take the positive root because we're told that the determinant is positive.
