A = (x - 3)/(2x - x ^ 2) / ((2 + x)/(2 - x) - (2 - x)/(2 + x) - (4x ^ 2)/(x ^ 2 - 4))

To simplify the expression A, we need to find a common denominator and combine the fractions in the numerator and denominator.

A = ((x - 3)/(2x - x^2)) / (((2 + x)/(2 - x)) - ((2 - x)/(2 + x)) - ((4x^2)/(x^2 - 4)))

First, let's simplify the fractions in the numerator and denominator:

A = ((x - 3)/(x(2 - x))) / (((2 + x)/(2 - x)) - ((2 - x)/(2 + x)) - ((4x^2)/((x + 2)(x - 2))))

Now, let's find a common denominator for the fractions in the denominator:

A = ((x - 3)/(x(2 - x))) / ((((2 + x)(2 + x))/((2 - x)(2 + x))) - (((2 - x)(2 - x))/((2 + x)(2 - x))) - ((4x^2)/((x + 2)(x - 2))))

A = ((x - 3)/(x(2 - x))) / (((4 + 4x + x^2 - 4 + 4x - x^2)/((2 - x)(2 + x))) - ((4 - 4x - 4 + x - 4x + x^2)/((2 - x)(2 + x))) - ((4x^2)/((x + 2)(x - 2))))

A = ((x - 3)/(x(2 - x))) / ((8x)/((2 - x)(2 + x)) - (2x)/((2 - x)(2 + x)) - ((4x^2)/((x + 2)(x - 2))))

Now, let's combine the fractions in the denominator:

A = ((x - 3)/(x(2 - x))) / ((8x - 2x - 4x^2)/((2 - x)(2 + x)(x + 2)(x - 2)))

A = ((x - 3)/(x(2 - x))) / ((6x - 4x^2)/((2 - x)(2 + x)(x + 2)(x - 2)))

A = ((x - 3)/(x(2 - x))) / ((2x(3 - 2x))/((2 - x)(2 + x)(x + 2)(x - 2)))

Therefore, the simplified expression for A is:

A = (x - 3)/(2x(3 - 2x))