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From my knowledge of stability, I understand that if the function approaches a finite number then the system will be stable. Thus if a system is stable its poles will be on the left of the $j\omega$-axis. This means that $Re(p_{1}, p_{2}) < 0 $. What I don't understand is how you determine whether the Imaginary part is equal to 0 or not.

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If you have complex conjugate poles (i.e., $\text{Im}\{p_i\}\neq 0$) you have oscillations in the step response. Only real-valued poles will give you a monotonic step response as shown in the figure.

Take a look at this related answer.

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