If z=1+√2i the find the value of the expression z⁴-4z³+10z^​​2-12z+10

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$$\begin{gathered} \boxed{\mathrm{z}=1+\sqrt{2}\mathrm{i}} \\ \mathrm{Square} \mathrm{both} \mathrm{sides} \\ {\mathrm{z}}^2={\left(1+\sqrt{2}\mathrm{i}\right)}^2=1+{\left(\sqrt{2}\mathrm{i}\right)}^2+2\sqrt{2}\mathrm{i}=1+2{\mathrm{i}}^2+2\sqrt{2}\mathrm{i}=1-2+2\sqrt{2}\mathrm{i}= \\ \Rightarrow \boxed{{\mathrm{z}}^2=-1+2\sqrt{2}\mathrm{i}} \\ \Rightarrow {\mathrm{z}}^3={\mathrm{z}}^2.\mathrm{z} \\ =\left(-1+2\sqrt{2}\mathrm{i}\right)\left(1+\sqrt{2}\mathrm{i}\right) \\ =-1-\sqrt{2}\mathrm{i}+2\sqrt{2}\mathrm{i}+4{\mathrm{i}}^2 \\ =-1+\sqrt{2}\mathrm{i}-4 \\ \boxed{{\mathrm{z}}^3=-5+\sqrt{2}\mathrm{i}} \\ {\mathrm{z}}^4={\left({\mathrm{z}}^2\right)}^2 \\ ={\left(-1+2\sqrt{2}\mathrm{i}\right)}^2 \\ =1+8{\mathrm{i}}^2-4\sqrt{2}\mathrm{i} \\ =1-8-4\sqrt{2}\mathrm{i} \\ \boxed{{\mathrm{z}}^4=-7-4\sqrt{2}\mathrm{i}} \\ \mathrm{Now} \mathrm{put} \mathrm{all} \mathrm{the} \mathrm{values} \mathrm{in} \mathrm{required} \mathrm{expression}. \end{gathered}$$

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