$\frac{ 2{x}^{3}-2{x}^{2}+{x}^{2}-4x+3 }{ {x}^{2}-2x+1 }$
Write $-4x$ as a difference$\frac{ 2{x}^{3}-2{x}^{2}+{x}^{2}-x-3x+3 }{ {x}^{2}-2x+1 }$
Use ${a}^{2}-2ab+{b}^{2}={\left( a-b \right)}^{2}$ to factor the expression$\frac{ 2{x}^{3}-2{x}^{2}+{x}^{2}-x-3x+3 }{ {\left( x-1 \right)}^{2} }$
Factor out $2{x}^{2}$ from the expression$\frac{ 2{x}^{2} \times \left( x-1 \right)+{x}^{2}-x-3x+3 }{ {\left( x-1 \right)}^{2} }$
Factor out $x$ from the expression$\frac{ 2{x}^{2} \times \left( x-1 \right)+x \times \left( x-1 \right)-3x+3 }{ {\left( x-1 \right)}^{2} }$
Factor out $-3$ from the expression$\frac{ 2{x}^{2} \times \left( x-1 \right)+x \times \left( x-1 \right)-3\left( x-1 \right) }{ {\left( x-1 \right)}^{2} }$
Factor out $x-1$ from the expression$\frac{ \left( x-1 \right) \times \left( 2{x}^{2}+x-3 \right) }{ {\left( x-1 \right)}^{2} }$
Cancel out the common factor $x-1$$\frac{ 2{x}^{2}+x-3 }{ x-1 }$
Write $x$ as a difference$\frac{ 2{x}^{2}+3x-2x-3 }{ x-1 }$
Factor out $x$ from the expression$\frac{ x \times \left( 2x+3 \right)-2x-3 }{ x-1 }$
Factor out the negative sign from the expression$\frac{ x \times \left( 2x+3 \right)-\left( 2x+3 \right) }{ x-1 }$
Factor out $2x+3$ from the expression$\frac{ \left( 2x+3 \right) \times \left( x-1 \right) }{ x-1 }$
Cancel out the common factor $x-1$$2x+3$