The missing polynomial must be of degree 1, since the dividend has degree 4 and the divisor has degree 3. So we're looking for a polynomial [tex]ax+b[/tex] such that
[tex](x^3-3)(ax+b)=x^4+5x^3-3x-15[/tex]
Expanding the left hand side gives
[tex]ax^4+bx^3-3ax-3b[/tex]
We need to have
[tex]\begin{cases}ax^4=x^4\\bx^3=5x^3\\-3ax=-3x\\-3b=-15\implies a=1,b=5[/tex]
so that the missing polynomial is [tex]\boxed{x+5}[/tex].
