$\frac{ 2{x}^{4}+{x}^{3}-8{x}^{2}+x+3 }{ 2x-3 }$
Write ${x}^{3}$ as a sum$\frac{ 2{x}^{4}-3{x}^{3}+4{x}^{3}-8{x}^{2}+x+3 }{ 2x-3 }$
Write $-8{x}^{2}$ as a difference$\frac{ 2{x}^{4}-3{x}^{3}+4{x}^{3}-6{x}^{2}-2{x}^{2}+x+3 }{ 2x-3 }$
Write $x$ as a difference$\frac{ 2{x}^{4}-3{x}^{3}+4{x}^{3}-6{x}^{2}-2{x}^{2}+3x-2x+3 }{ 2x-3 }$
Factor out ${x}^{3}$ from the expression$\frac{ {x}^{3} \times \left( 2x-3 \right)+4{x}^{3}-6{x}^{2}-2{x}^{2}+3x-2x+3 }{ 2x-3 }$
Factor out $2{x}^{2}$ from the expression$\frac{ {x}^{3} \times \left( 2x-3 \right)+2{x}^{2} \times \left( 2x-3 \right)-2{x}^{2}+3x-2x+3 }{ 2x-3 }$
Factor out $-x$ from the expression$\frac{ {x}^{3} \times \left( 2x-3 \right)+2{x}^{2} \times \left( 2x-3 \right)-x \times \left( 2x-3 \right)-2x+3 }{ 2x-3 }$
Factor out the negative sign from the expression$\frac{ {x}^{3} \times \left( 2x-3 \right)+2{x}^{2} \times \left( 2x-3 \right)-x \times \left( 2x-3 \right)-\left( 2x-3 \right) }{ 2x-3 }$
Factor out $2x-3$ from the expression$\frac{ \left( 2x-3 \right) \times \left( {x}^{3}+2{x}^{2}-x-1 \right) }{ 2x-3 }$
Cancel out the common factor $2x-3$${x}^{3}+2{x}^{2}-x-1$