1. **State the problem:** Simplify the expressions involving square roots: $\sqrt{143}$, $7\sqrt{105}$, $6\sqrt{105}$, and $210\sqrt{3}$.\n\n2. **Recall the simplification rule:** For square roots, if the radicand (number inside the root) can be factored into a perfect square times another number, we can simplify as $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.\n\n3. **Simplify $\sqrt{143}$:**\n$143 = 11 \times 13$, neither 11 nor 13 is a perfect square, so $\sqrt{143}$ cannot be simplified further.\n\n4. **Simplify $7\sqrt{105}$:**\nFactor 105: $105 = 3 \times 5 \times 7$, no perfect square factors, so $7\sqrt{105}$ remains as is.\n\n5. **Simplify $6\sqrt{105}$:**\nSame as above, $6\sqrt{105}$ remains as is.\n\n6. **Simplify $210\sqrt{3}$:**\n$210 = 2 \times 3 \times 5 \times 7$, no perfect square factors, so $210\sqrt{3}$ remains as is.\n\n**Final answers:**\n- $\sqrt{143}$ (cannot simplify)\n- $7\sqrt{105}$ (cannot simplify)\n- $6\sqrt{105}$ (cannot simplify)\n- $210\sqrt{3}$ (cannot simplify)