We divide by long division method
#" " " " " " " underline(x^2+6x+9/2)#
#2x-3|~2x^3+9x^2-9x+1#
#" " " " " "underline(2x^3-3x^2" " " " " " " ")#
#" " " " " " " " " "12x^2-9x+1#
#" " " " " " " " " "underline(12x^2-18x" " " )#
#" " " " " " " " " " " " " " "+9x+1#
#" " " " " " " " " " " " " "underline(+9x-27/2" " " )#
#" " " " " " " " " " " " " " " " " " "+29/2#
The result is
#(2x^3+9x^2-9x+1)/(2x-3)=x^2+6x+9/2+(29/2)/(2x-3)#
Checking:
#"Divisor x Quotient"+"Remainder"="Dividend"#
#(2x-3)(x^2+6x+9/2)+29/2=2x^3+12x^2+9x-3x^2-18x-27/2+29/2#
#(2x-3)(x^2+6x+9/2)+29/2=2x^3+9x^2-9x-27/2+29/2#
#(2x-3)(x^2+6x+9/2)+29/2=2x^3+9x^2-9x+2/2#
#(2x-3)(x^2+6x+9/2)+29/2=2x^3+9x^2-9x+1#
God bless....I hope the explanation is useful.
