1. **State the problem:** We need to find the remainder when the polynomial $p(x) = 15x^3 + 22x^2 - 15x + 2$ is divided by $x + 1$. 2. **Recall the Remainder Theorem:** The remainder when a polynomial $p(x)$ is divided by $x - a$ is $p(a)$. 3. **Apply the theorem:** Here, the divisor is $x + 1$, which can be written as $x - (-1)$, so $a = -1$. 4. **Evaluate $p(-1)$:** $$p(-1) = 15(-1)^3 + 22(-1)^2 - 15(-1) + 2$$ $$= 15(-1) + 22(1) + 15 + 2$$ $$= -15 + 22 + 15 + 2$$ $$= 24$$ 5. **Conclusion:** The remainder when $p(x)$ is divided by $x + 1$ is $24$.