Multiplication of two factors with complex numbers

I have the following to multiply ;

$$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$

Now, I think that the product must not have any complex numbers...

But here is what I get

$$z^2-2zp+p^2+q^1+\sqrt{t^2+u^2}+(\sqrt{t-ui}+\sqrt{t+ui})(z-p+qi)$$

I'm pretty sure that there is some step that I'm missing... Can anybody point me out my error ? Thanks

• why are you assuming that the answer will not have any complex numbers? In fact the two numbers you are multiplying need not be complex conjugates (because of the presence of $z$). – Anurag A Apr 21 '15 at 16:35
• For no particular reason, I just thought that they needed to go... – user108343 Apr 21 '15 at 16:36

I think you made a mistake about sign. See the red sign at the last. $$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$ $$=z^2+z(-p+qi)+z\sqrt{t-ui}+z(-p-qi)+(-p-qi)(-p+qi)+(-p-qi)\sqrt{t-ui}+z\sqrt{t+ui}+(-p+qi)\sqrt{t+ui}+\sqrt{(t+ui)(t-ui)}$$ $$=z^2+z(-p+qi-p-qi)+z(\sqrt{t-ui}+\sqrt{t+ui})+p^2+q^2+(-p-qi)\sqrt{t-ui}+(-p+qi)\sqrt{t+ui}+\sqrt{t^2+u^2}$$ $$=z^2-2pz+z(\sqrt{t-ui}+\sqrt{t+ui})+p^2+q^2+(-p-qi)\sqrt{t-ui}+(-p+qi)\sqrt{t+ui}+\sqrt{t^2+u^2}$$ $$=z^2-2pz+p^2+q^2+\sqrt{t^2+u^2}+(z-p)(\sqrt{t-ui}+\sqrt{t+ui})+qi(\sqrt{t+ui}\color{red}{-}\sqrt{t-ui})$$