To rationalize the expression, we have to find common terms between numerators and denominators and then simplify:
[tex]\frac{2m^2-4m}{2\cdot\left(m-2\right)}=\frac{2m\cdot(m-2)}{2\cdot(m-2)}[/tex][tex]=\frac{2}{2}\cdot m\cdot\frac{(m-2)}{(m-2)}=\frac{1}{1}\cdot m\cdot\frac{1}{1}=m[/tex]To simplify the numerator of the second expression we have to remember that:
[tex]x^2-2ax+a^2=(x-a)^2[/tex]Then, the expression would be simplified as:
[tex]\frac{m^2-2m+1}{m-1}=\frac{(m-1)^2}{m-1}[/tex][tex]=\frac{(m-1)^2}{m-1}=(m-1)\cdot\frac{m-1}{m-1}=(m-1)\cdot1=m-1[/tex]For the third expression, when factorizing we have to find two numbers that when added equals -3 and when multiplied equals +2, meaning:
[tex]x^2+(a+b)x+a\cdot b=(x+a)\cdot(x+b)[/tex]Then, the factorization of the numerator of the third expression is:
[tex]\frac{m^2-3m+2}{m^2-m}=\frac{(m-1)(m-2)}{m(m-1)}[/tex][tex]=\frac{(m-1)}{(m-1)}\cdot\frac{(m-2)}{m}=1\cdot\frac{(m-2)}{m}=\frac{m-2}{m}[/tex]Finally, to simplify the numerator of the last expression we will use the same factorization as in the third one. And for the denominator we have to use the following:
[tex](x^2-a^2)=(x-1)(x+1)[/tex][tex]\frac{m^2-m-2}{m^2-1}=\frac{(m+1)(m-2)}{(m_{}+1)(m-1)}[/tex][tex]=\frac{(m+1)}{(m_{}+1)}\cdot\frac{(m-2)}{(m-1)}=1\cdot\frac{(m-2)}{(m-1)}=\frac{m-2}{m-1}[/tex]Answer:
[tex]\frac{m-2}{m}=\frac{m^2-3m+2}{m^2-m}[/tex][tex]m-1=\frac{m^2-2m+1}{m-1}[/tex][tex]\frac{m-2}{m-1}=\frac{m^2-m-2}{m^2-1}[/tex][tex]m=\frac{2m^2-4m}{2\cdot(m-2)}[/tex]