1. **State the problem:** Simplify the expression $(2u - 3v - 1) - (u + 3v + u)$. 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to each term inside the second parentheses. $$ (2u - 3v - 1) - (u + 3v + u) = 2u - 3v - 1 - u - 3v - u $$ 3. **Combine like terms:** Group the terms with $u$, $v$, and constants separately. $$ (2u - u - u) + (-3v - 3v) + (-1) $$ 4. **Simplify each group:** $$ 2u - u - u = \cancel{2u} - \cancel{u} - u = 0u = 0 $$ $$ -3v - 3v = -6v $$ $$ -1 = -1 $$ 5. **Write the simplified expression:** $$ 0 - 6v - 1 = -6v - 1 $$ **Final answer:** $$ -6v - 1 $$