Evaluate: (2x^4+x^3-8x^2+x+3)/(2x-3)

Expression: $\left( 2{x}^{4}+{x}^{3}-8{x}^{2}+x+3 \right)\div\left( 2x-3 \right)$

Write the division as a fraction

$\frac{ 2{x}^{4}+{x}^{3}-8{x}^{2}+x+3 }{ 2x-3 }$

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 $2x-3$

${x}^{3}+2{x}^{2}-x-1$