Let f(x) be the required polynomial.
Then, [tex]\frac{x^{4}+5x^{3}-3x-15}{f(x)} =x^{3} -3[/tex]
Or, [tex]f(x) = \frac{x^{4}+15x^{3}-3x-15}{x^{3}-3 }[/tex]
Consider the numerator.
[tex]x^{4} +5x^{3} -3x-15 = x^{3}(x + 5)-3(x + 5)[/tex]
[tex]=(x^{3}- 3)(x+5)[/tex]
Therefore, [tex]\frac{x^{4}+5x^{3}-3x-15}{x^{3}-3 }[/tex]
= [tex]\frac{(x^{3}-3)(x+5) }{x^{3}-3 }[/tex]
= x + 5
