Answer :
We are given the division
$$
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}.
$$
To find the quotient, we perform polynomial long division as follows.
1. **Divide the leading terms:**
The leading term of the dividend is $x^4$ and the leading term of the divisor is $x^3$. Dividing these gives
$$
\frac{x^4}{x^3} = x.
$$
2. **Multiply and subtract:**
Multiply the divisor by $x$:
$$
x(x^3 - 3) = x^4 - 3x.
$$
Subtract this product from the dividend:
$$
\begin{aligned}
&\quad \; \left(x^4 + 5x^3 - 3x - 15\right) \\
&- \left(x^4 - 3x\right) \\
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$
3. **Repeat the division:**
Now, the new dividend is $5x^3 - 15$. Divide its leading term, $5x^3$, by the leading term of the divisor, $x^3$:
$$
\frac{5x^3}{x^3} = 5.
$$
4. **Multiply and subtract again:**
Multiply the divisor by $5$:
$$
5(x^3 - 3) = 5x^3 - 15.
$$
Subtract this from $5x^3 - 15$:
$$
\begin{aligned}
&\quad \; \left(5x^3 - 15\right) \\
&- \left(5x^3 - 15\right) \\
&= 0.
\end{aligned}
$$
Since the remainder is $0$, the division is exact. The quotient combined is the sum of the terms we obtained, which is
$$
x + 5.
$$
Thus, the quotient of $\left(x^4+5 x^3-3 x-15\right)$ divided by $\left(x^3-3\right)$ is
$$
\boxed{x+5}.
$$
$$
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}.
$$
To find the quotient, we perform polynomial long division as follows.
1. **Divide the leading terms:**
The leading term of the dividend is $x^4$ and the leading term of the divisor is $x^3$. Dividing these gives
$$
\frac{x^4}{x^3} = x.
$$
2. **Multiply and subtract:**
Multiply the divisor by $x$:
$$
x(x^3 - 3) = x^4 - 3x.
$$
Subtract this product from the dividend:
$$
\begin{aligned}
&\quad \; \left(x^4 + 5x^3 - 3x - 15\right) \\
&- \left(x^4 - 3x\right) \\
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$
3. **Repeat the division:**
Now, the new dividend is $5x^3 - 15$. Divide its leading term, $5x^3$, by the leading term of the divisor, $x^3$:
$$
\frac{5x^3}{x^3} = 5.
$$
4. **Multiply and subtract again:**
Multiply the divisor by $5$:
$$
5(x^3 - 3) = 5x^3 - 15.
$$
Subtract this from $5x^3 - 15$:
$$
\begin{aligned}
&\quad \; \left(5x^3 - 15\right) \\
&- \left(5x^3 - 15\right) \\
&= 0.
\end{aligned}
$$
Since the remainder is $0$, the division is exact. The quotient combined is the sum of the terms we obtained, which is
$$
x + 5.
$$
Thus, the quotient of $\left(x^4+5 x^3-3 x-15\right)$ divided by $\left(x^3-3\right)$ is
$$
\boxed{x+5}.
$$
