Lets divide 3t4+5t3−7t2+2t+2 by t2+3t+1,
3t2−4t+2t2+3t+1 )¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 3t4+5t3−7t2+2t+2 −3t4±9t3±3t2––––––––––––––––– −4t3−10t2+2t∓4t3∓12t2∓4t––––––––––––––––––– 2t2+6t+2−2t2±6t±2–––––––––––––– 00
So the quotient, q(x) = 3t^2 - 4t + 2 and the remainder, r(x) = 0