if a is a root, then f(a) = 0

by inspection of the coefficients (they add to 0), then 1 is a root and x - 1 is a factor(note that f(1) = 2 + 1 - 5 + 2 = 0

divide either synthetically or by long division to get:

1 | ... 2 .... 1 ..... -5 ..... 2...... ......... 2..... 3 .....-2...---------------- ---------------- (add)....... 2...... 3 ......-2 .....0 <== remainder is 0, confirming that x - 1 is a factor

quotient is 2x^2 + 3x - 2, which factors as (2x - 1)(x + 2)

thus, fully factored this is f(x) = 2x^3 + x^2 - 5x + 2 = (x - 1)(2x - 1)(x + 2)

................x^2 ..+ 2x .+ 1/2.........------------------------------...2x - 3 | ... 2x^3 + x^2 - 5x + 2...............2x^3 - 3x^2............. ------------------ (subtract)........................ 4x^2 - 5x....................... 4x^2 - 6x........................ ------------- (subtract)...................................x + 2........................... .......x - 1.5.................................. ---------............................ ........... 7/2 <== remainder

(2x^3 + x^2 - 5x + 2) / (2x - 3) = x^2 + 2x + 1/2 + (7/2) / (2x - 3)

or, 2x^3 + x^2 - 5x + 2 = (x^2 + 2x + 1/2)(2x - 3) + 7/2

just looking at the constants: (1/2)(-3) + 7/2 = -3/2 + 7/2 = 2 as needed

remainder = 7/2