Which line will have no solution with the parabola y – x + 2 = x2? y = –3x –3

You list only one line:  y = -3x - 3, so that 's all we can check.

If the parabola is represented by   y – x + 2 = x^2, then replace y here by 
y = -3x - 3:

-3x - 3  – x + 2 = x^2.  Then determine whether or not this new equation has a real solution:

Combining like terms,       -3x - x -3 + 2 = x^2, or

                                            -4x -1 = x^2, or       x^2 + 4x + 1 = 0.

Does this have real solutions?  Find the determinant, b^2 - 4ac:
                       
                        
determinant = (4)^2 - 4(1)(1) = 16 - 4 = 12

Thus, this quadratic equation will have two real, unequal roots.  At least one of these roots should be the x-coordinate of the point of intersection(s) of the given parabola and line.

You'll need to check out your other answer choices using the same or a similar method.