`\frac{24x^3-18x^2-25x+9}{12x+9}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{3(8x^3-6x^2-5x+3)}{3(4x+3)}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{8x^3-8x^2+2x^2-2x-3x+3}{4x+3}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{8x^2(x-1)+2x(x-1)-3(x-1)}{4x+3}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{(x-1)(8x^2+2x-3)}{4x+3}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{(x-1)(8x^2+6x-4x-3)}{4x+3}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{(x-1).[ 2x(4x+3)-(4x+3)] }{4x+3}+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=(x-1)(2x-1)+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=2x^2-x-2x+1+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=2x^2-3x+1+\frac{x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{2x^2(x^2-2x+3)-3x(x^2-2x+3)+x^2-2x+3+x^3-7x^2+11x-3}{x^2-2x+3}`
`=\frac{2x^4-4x^3+6x^2-3x^3+6x^2-9x+x^2-2x+x^3-7x^2+11x}{x^2-2x+3}`
`=\frac{2x^4-6x^3+6x^2}{x^2-2x+3}`
