There is an error in the question. The polynomial should be:
2x4-5x3+2x2-x+2.
To avoid actual division we have to assume factorisation has to be used.
x2-3x+2=(x-1)(x-2). By substituting x=1 and x=2 into the polynomial, we find that it evaluates to zero in each case: 2-5+2-1+2=0 and 32-40+8-2+2=0. Therefore the polynomial is divisible exactly by the divisor and can be written:
(x-1)(x-2)(ax2+bx+c) where a, b and c are constants.
Since the leading coefficient is 2, then a×1×1=2, making a=2; the constant is 2 so c×1×2=2, and c=1.
(x2-3x+2)(2x2+bx+1)=2x4-5x3+2x2-x+2.
To find b we need to look at all the x terms created by the product.
We have -3x+2bx=-x, so -3+2b=-1, 2b=-1+3=2, b=1.
So the result of the division has to be 2x2+x+1.