(5x+3/x)-(x-1/2x)

(5x+3/x)-(x-1/2x) :

In mathematics the first bracket is only decorative.

(5x+3/x) =5x+(3/x). And cannot mean (5x+3)/x.

-(x-1/2x) = -x+(1/2)x, and cannot mean (x-1)/(2x). 1/2x is not 1/(2x) but it means (1/2)x.

Therefore

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

=(5-1+0.5)x+3/x = 4.5x+3/x

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If you wanted the factors:The expression is equal to  (1/x)(4.5x^2 +3)  or  (2/x)(9x^2 +6) .

If you wanted to solve for zeros:

No real solution and Y axis is an asymptote.

y=4.5x is another obleque asynptote.

The expression is discontinuous at x= 0 with an infinite jump from positive infinity to negative infinity as x change sign around zero i.e from  0+  to  0- .