a) To simplify the expression x^2 / 3x+6 + 4x+4 / 3x+6, we need to find a common denominator for the two fractions. The common denominator is (3x+6).

Now, we can rewrite the expression as (x^2 + 4x + 4) / (3x+6).

Next, we can factor the numerator: (x+2)(x+2) / (3x+6).

Finally, we can cancel out the common factors: (x+2)(x+2) / 3(x+2).

The simplified expression is (x+2) / 3.

b) To simplify the expression 2/x + 3/x+1, we need to find a common denominator for the two fractions. The common denominator is x(x+1).

Now, we can rewrite the expression as (2(x+1) + 3x) / x(x+1).

Simplifying further, we get (2x + 2 + 3x) / x(x+1).

Combining like terms, we have (5x + 2) / x(x+1).

c) To simplify the expression x/x-3 + 5/x+3, we need to find a common denominator for the two fractions. The common denominator is (x-3)(x+3).

Now, we can rewrite the expression as (x(x+3) + 5(x-3)) / (x-3)(x+3).

Expanding and simplifying the numerator, we get (x^2 + 3x + 5x - 15) / (x-3)(x+3).

Combining like terms, we have (x^2 + 8x - 15) / (x-3)(x+3).

d) To simplify the expression x + 1/2x-2 + -2x/x^2-1, we need to find a common denominator for the three fractions. The common denominator is (2x-2)(x+1)(x-1).

Now, we can rewrite the expression as (x(x+1)(x-1) + (2x-2) + (-2x(x+1))) / (2x-2)(x+1)(x-1).

Expanding and simplifying the numerator, we get (x^3 + x^2 - x + 2x - 2 - 2x^2 - 2x) / (2x-2)(x+1)(x-1).

Combining like terms, we have (x^3 - x^2 - x - 2) / (2x-2)(x+1)(x-1).