1. **State the problem:** Multiply the polynomials $\left(x^3 + x^2 + 1\right)$ and $\left(x^2 - x - 5\right)$.\n\n2. **Recall the distributive property:** To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.\n\n3. **Multiply each term:**\n$$\begin{aligned} &x^3 \cdot x^2 = x^{3+2} = x^5 \\ &x^3 \cdot (-x) = -x^{3+1} = -x^4 \\ &x^3 \cdot (-5) = -5x^3 \\ &x^2 \cdot x^2 = x^{2+2} = x^4 \\ &x^2 \cdot (-x) = -x^{2+1} = -x^3 \\ &x^2 \cdot (-5) = -5x^2 \\ &1 \cdot x^2 = x^2 \\ &1 \cdot (-x) = -x \\ &1 \cdot (-5) = -5 \end{aligned}$$\n\n4. **Write all terms together:**\n$$x^5 - x^4 - 5x^3 + x^4 - x^3 - 5x^2 + x^2 - x - 5$$\n\n5. **Combine like terms:**\n- Combine $-x^4$ and $x^4$ to get $\cancel{-x^4} + \cancel{x^4} = 0$.\n- Combine $-5x^3$ and $-x^3$ to get $-6x^3$.\n- Combine $-5x^2$ and $x^2$ to get $-4x^2$.\n\nSo the simplified polynomial is:\n$$x^5 - 6x^3 - 4x^2 - x - 5$$\n\n**Final answer:**\n$$\boxed{x^5 - 6x^3 - 4x^2 - x - 5}$$