[tex]\displaystyle\\\bf\\
z_1=1-m+i\\
z_2=m+1-2mi\\\\
z_1\cdot z_2=(1-m+i)(m+1-2mi)=\\
=m-m^2+mi+1-m+i-2mi+2m^2i-2mi^2=~~(dar~i^2=-1)\\
=m-m^2+mi+1-m+i-2mi+2m^2i+2m=\\
=-m^2+2m+1-mi+i+2m^2i=\\
= -m^2+2m+1+(2m^2-m+1)\cdot i\\\\
z_1\cdot z_2=-m^2+2m+1+(2m^2-m+1)\cdot i\\\\
z_1\cdot z_2\in R~~daca~~(2m^2-m+1)=0\\
~~~~~~unde~~~(2m^2-m+1)~este~coeficientul~lui~i.\\\\
Rezolvam~ecuatia:\\\\
2m^2-m+1=0\\
\Delta=(-1)^2-4\cdot 2\cdot 1 = 1 - 9 = -8\\
\Delta\ \textless \ 0\\
\Rightarrow~m\notin R \\
m \in \Phi[/tex]
⇒ Nu exista nicio valoare reala pentru m astfel incat z₁·z₂ ∈ R.
