Prove that 2x - 2/( x^2 + 3 ) = f(x) if f(x) = ( 2x^3 + 6x - 2 )/ ( x^2 + 3 )

To prove that 2x - 2/( x^2 + 3 ) = f(x), we'll substitute
f(x) by the given expression ( 2x^3 + 6x - 2 )/ ( x^2 + 3 ).We could manage the problem
in 2 ways: either we'll calculate the left side, namely the difference 2x - 2/( x^2 + 3
), or we'll re-write the expression of f(x).

We'll re-write
f(x):

( 2x^3 + 6x - 2 )/ (x^2 + 3) = (2x^3 + 6x)/(x^2 + 3)
- 2/(x^2 + 3)

We'll factorize by 2x the first
ratio:

(2x^3 + 6x)/(x^2 + 3) = 2x(x^2 + 3)/(x^2 +
3)

We'll simplify the first
ratio:

2x(x^2 + 3)/(x^2 + 3) =
2x

f(x) = 2x - 2/(x^2 +
3)

We notice that the LHS =
RHS, therefore 2x - 2/(x^2 + 3) = ( 2x^3 + 6x - 2 )/
(x^2 + 3).