The polynomial \(4x ^ {10} - x ^ 9 + 3x ^ {11} - 5x ^ 7 + c x ^ 6 + 2x ^ 5 - x ^ 4 + x ^ 3 - 4x ^ 2 + 6x - 2\) when divided by \(x - 1\) leaves a remainder \(2.\) Then the value of \(c + 6\) is

The value of $c+6$ is 5. The remainder theorem gives the remainder by substituting $x=1$, where the terms other than $cx^6$ sum to $3$. Thus $c+3=2$, so $c=-1$ and $c+6=5$. The value $-1$ is the coefficient alone.