Answer :
We want to divide the polynomial
$$
x^4 + 5x^3 - 3x - 15
$$
by
$$
x^3 - 3.
$$
**Step 1.**
Divide the leading term of the numerator, $x^4$, by the leading term of the denominator, $x^3$, to obtain the first term of the quotient:
$$
\frac{x^4}{x^3} = x.
$$
Multiply the entire denominator by $x$:
$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$
Subtract this from the original numerator:
\[
\begin{aligned}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
\]
**Step 2.**
Now, take the new polynomial $5x^3 - 15$ and divide its leading term $5x^3$ by the leading term of the denominator $x^3$:
$$
\frac{5x^3}{x^3} = 5.
$$
Multiply the denominator by $5$:
$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$
Subtract this product from $5x^3 - 15$:
$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$
Since the remainder is zero, the division finishes here.
**Conclusion.**
The quotient of the division is the sum of the terms obtained:
$$
x + 5.
$$
Thus, the answer is $\boxed{x+5}$.
$$
x^4 + 5x^3 - 3x - 15
$$
by
$$
x^3 - 3.
$$
**Step 1.**
Divide the leading term of the numerator, $x^4$, by the leading term of the denominator, $x^3$, to obtain the first term of the quotient:
$$
\frac{x^4}{x^3} = x.
$$
Multiply the entire denominator by $x$:
$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$
Subtract this from the original numerator:
\[
\begin{aligned}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
\]
**Step 2.**
Now, take the new polynomial $5x^3 - 15$ and divide its leading term $5x^3$ by the leading term of the denominator $x^3$:
$$
\frac{5x^3}{x^3} = 5.
$$
Multiply the denominator by $5$:
$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$
Subtract this product from $5x^3 - 15$:
$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$
Since the remainder is zero, the division finishes here.
**Conclusion.**
The quotient of the division is the sum of the terms obtained:
$$
x + 5.
$$
Thus, the answer is $\boxed{x+5}$.
