Let's first multiply out the left side of the equation:

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

Now let's multiply out the right side of the equation:

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

Now we can set the two sides equal to each other:

2x - 3x^2 - 3 = -x^2 + x

Rearranging terms, we get:

2x - 3x^2 - 3 + x^2 - x = 0
-x^2 - x - 3 = 0

This is a quadratic equation which can be solved using the quadratic formula:

x = (-(-1) ± √((-1)^2 - 4(-1)(-3))) / (2*(-1))
x = (1 ± √(1 + 12)) / -2
x = (1 ± √13) / -2

Therefore, the solution to the equation (1-3x)(x+1)=(1+x)(x-1) is:

x = (1 + √13) / -2 or x = (1 - √13) / -2