ExpandAll[
      Simplify[(d^2 + 2 d (1 - b) - 
                4 d (1 - b) c + (1 - b)^2 - 4 (1 - b)^2 c + 
                4 (1 - b)^2 c^2) 
            c (b - d + (1 - b) c) - (1 - 2 d - 2 (1 - b) + 
                4 (1 - b) c + d^2 + 2 d (1 - b) - 
                4 d (1 - b) c + (1 - b)^2 - 4 (1 - b)^2 c + 
                4 (1 - b)^2 c^2) (1 - 
                c) (d - (1 - b) c)]]

Solve[
      b c - (b^2) c + (c^2) - 3 b (c^2) + 2 (b^2) (c^2) - (b^2) d - c d - 
          2 b c d + 4 (b^2) c d - 2 (c^2) d + 6 b (c^2) d - 
          4 (b^2) (c^2) d + 2 b (d^2) + 3 c (d^2) - 4 b c (d^2) - 
          (d^3) == 0, c]