1. The problem asks to find the missing number in the equation $$4x - 10 = \_ x + 12$$ so that the equation has no solutions. 2. For a linear equation $$ax + b = cx + d$$ to have no solutions, the lines must be parallel and distinct. This means the coefficients of $$x$$ must be equal, but the constants must be different. 3. Here, the coefficient of $$x$$ on the left side is 4. Let the missing number be $$m$$, so the equation becomes: $$4x - 10 = mx + 12$$ 4. For no solutions, set the coefficients of $$x$$ equal: $$4 = m$$ 5. Check the constants: Left side constant is $$-10$$, right side constant is $$12$$, which are different. 6. Since $$m = 4$$ and constants differ, the equation has no solutions. Final answer: $$\boxed{4}$$