Solution Factor: x3+1=(x+1)(x2−x+1)x^3+1=(x+1)(x^2-x+1)x3+1=(x+1)(x2−x+1) Let: 3x2+3x3+1=Ax+1+Bx+Cx2−x+1\frac{3x^2+3}{x^3+1}=\frac{A}{x+1}+\frac{Bx+C}{x^2-x+1}x3+13x2+3=x+1A+x2−x+1Bx+C Solving gives: A=2,B=1,C=1A=2,\quad B=1,\quad C=1A=2,B=1,C=1 Hence: 3x2+3x3+1=2x+1+x+1x2−x+1\boxed{\frac{3x^2+3}{x^3+1}=\frac{2}{x+1}+\frac{x+1}{x^2-x+1}}x3+13x2+3=x+12+x2−x+1x+1