Use the polynomial remainder theorem: the remainder upon dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] is [tex]p(c)[/tex].
[tex](-b)^4+3(-b)^2-2(-b)+2=b^4+3b^2+2b+2[/tex]
[tex]((-b)^2-3)^2=b^4-6b^2+9[/tex]
Now
[tex]b^4+3b^2+2b+2=b^4-6b^2+9\implies9b^2+2b-7=0[/tex]
[tex]\implies(9b-7)(b+1)=0[/tex]
[tex]\implies b=\dfrac79\text{ or }b=-1[/tex]
