[tex](-3)^4+(-3)^2-2=88[/tex], so by the polynomial remainder theorem, the remainder is 88.
To verify this, we can compute the quotient remainder:
[tex]h^4=h^3\cdot h[/tex], and [tex]h^3(h+3)=h^4+3h^3[/tex]. Subtracting from the numerator gives a remainder of [tex]-3h^3+h^2-2[/tex]
[tex]-3h^3=-3h^2\cdot h[/tex], and [tex]-3h^2(h+3)=-3h^3-9h^2[/tex]. Subtracting from the previous remainder gives a new remainder of [tex]10h^2-2[/tex].
[tex]10h^2=10h\cdot h[/tex], and [tex]10h(h+3)=10h^2+30h[/tex]. Subtracting from the previous remainder gives a new remainder of [tex]-30h-2[/tex].
[tex]-30h=-30\cdot h[/tex], and [tex]-30(h+3)=-30h-90[/tex]. Subtracting from the previous remainder gives a new remainder of [tex]88[/tex], which doesn't contain factors of [tex]h[/tex], so we're done.
This means we have
[tex]\dfrac{h^4+h^2-2}{h+3}=h^3-3h^2+10h-30+\dfrac{88}{h+3}[/tex]
