The solution set of the equation x/x+3 + 1/x = 1-2x-3/x2 +3x is?

To solve the equation x/(x+3) + 1/x = 1 - 2x - 3/x^2 + 3x, we first need to find a common denominator for the fractions on the left side of the equation.

The common denominator for x/(x+3) and 1/x is x(x+3). So, we rewrite the equation as:

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

Now, we can combine the fractions on the left side:

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

Next, we multiply both sides by x(x+3) to get rid of the denominators:

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

Expanding and simplifying both sides, we get:

x^2 + x + 3 = x^2 + 3x - 2x^3 - 6x^2 - 3x - 9 + 3x^3 + 9x^2

Rearranging terms, we get:

0 = 0

This equation is always true, which means that the original equation is an identity and holds for all values of x. Therefore, the solution set of the equation x/(x+3) + 1/x = 1 - 2x - 3/x^2 + 3x is all real numbers.

The equation can be rewritten as:

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

Multiplying both sides by (x^3 - 1 / x):

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

Expanding and simplifying:

x = x^3 - 1 / x - 2x - 3 / x^2 - 3x

x = x^3 - 2x - 3 / x^2 - 3x - 1 / x

Combining like terms:

x = x^3 - 5x - 3 / x^2 - 1 / x

Factoring the numerator:

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

Expanding:

x = x^3 + 3x^2 + x - 3x^2 - 9x - 3 - 3 / x^2 - 1 / x

Combining like terms:

x = x^3 - 5x - 3 - 3 / x^2 - 1 / x

Now, we can see that the equation has two possible solutions:

x = 3

x = -1/3

Therefore, the solution set is {3, -1/3}.