Question

(2x3+x2-5x-2) by 2x+3

Answer 1

you need to find a root of the equation as this will always give you a factor. You do this by inputting values of x into the equation until you find f(x) =0 as this x value is a root of the eqation. In general if a is a root of an equation then(x-a) is a factor. 
here we get lucky straight off ie f(x) = 2x^3 + x^2 -5x +2 
and f(1) = 2 +1 - 5 + 2 = 0 
so x = 1 is a root and ( x - 1) is a factor and will therefore divide into f(x) with no remainder 

2x^2 + 3x - 2 this is dividing f(x) by ( x - 1) 
( x - 1 )}2x^3 +x^2 -5x +2 
2x^3 -2x^2 
------------- 
3x^2 -5x 
3x^2 - 3x 
-------------- 
-2x +2 
-2x + 2 

so we have f(x) = 2x^3 + x^2 -5x + 2 = (x - 1) ( 2x^2 +3x -2 ) 
but the quadratic can be factorised ie 2x^2 +3x - 2 = (2x - 1)(x + 2) 
so f(x) = (x - 1)( x+ 2)(2x -1) 
Now we have to divide f(x) by ( 2x -3) in the same way as we divided by ( x - 1) above 

x^2 + 2x 
----------- 
(2x-3) } 2x^3 + x^2 -5x + 2 
2x^3 - 3x^2 
-------------- 
4x^2 -5x 
4x^2 -6x 
-------------- 
x + 2 we can divide no further so we have the remainder 
which is ( x + 2) / (2x - 3) 
you can check this by multipling out (2x - 3)[ x^2 + 2x +( x +2)/(2x -3)] this should give you f(x) if not i've slipped up