The differential equation in the Question is ambiguously written. Your solution for the impulse response assumed that the differential equation is really:
D2x(t) + 2Dx(t) + 5x(t) = 5u(t), u(t) = delta(t), x(0-) = Dx(0-) = 0.
where u(t) is the generic symbol for the input to the system, not the unit step. But, the equation as written doesn't show the "u(t)" explicitly on the right hand side. Based on that, and the context of the question about overshoot, the fact that the question doesn't explicitly define the input but needs to be solved anyway, and that some treatments might be a bit loose in their notation for step inputs, I'm going to guess that the differential equation is really:
D2x(t) + 2Dx(t) + 5x(t) = u(t), u(t) = 5*step(t), x(0-) = Dx(0-) = 0.
where step(t) is the unit step function. If my guess is correct, you do really want that second form of Z(s), though more formally it would look like:
H(s) = 1/(s^2 + 2s + 5) % transfer function
U(s) = 5/s % Laplace of 5*step(t)
Z(s)= H(s)U(s) = (1/s^2+2s+5)*(5/s);
