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).

The first fraction x^2 / (3x+6) remains the same.

For the second fraction (4x+4) / (3x+6), we can simplify it by factoring out a common factor of 4 from the numerator and denominator: 4(x+1) / (3x+6).

Now we can combine the two fractions: x^2 / (3x+6) + 4(x+1) / (3x+6).

Since the fractions have the same denominator, we can add the numerators: (x^2 + 4(x+1)) / (3x+6).

To simplify further, we can distribute the 4: (x^2 + 4x + 4) / (3x+6).

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).

The first fraction 2/x remains the same.

For the second fraction 3/(x+1), we can multiply the numerator and denominator by x to get 3x / (x(x+1)).

Now we can combine the two fractions: 2/x + 3x / (x(x+1)).

Since the fractions have the same denominator, we can add the numerators: (2 + 3x) / (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).

The first fraction x/(x-3) remains the same.

For the second fraction 5/(x+3), we can multiply the numerator and denominator by (x-3) to get 5(x-3) / (x+3)(x-3).

Now we can combine the two fractions: x/(x-3) + 5(x-3) / (x+3)(x-3).

Since the fractions have the same denominator, we can add the numerators: (x + 5(x-3)) / (x+3)(x-3).

To simplify further, we can distribute the 5: (x + 5x - 15) / (x+3)(x-3).

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

The first term x remains the same.

For the second term 1/(2x-2), we can multiply the numerator and denominator by (x+1) to get (x+1) / (2x-2)(x+1).

For the third term -2x/(x^2-1), we can factor out a common factor of -2x from the numerator and denominator: -2x(x) / (x^2-1).

Now we can combine the three terms: x + (x+1) / (2x-2)(x+1) + -2x(x) / (x^2-1).

Since the terms have the same denominator, we can add the numerators: (x(2x-2)(x+1) + (x+1) - 2x(x)) / (2x-2)(x+1)(x^2-1).

To simplify further, we can distribute and combine like terms in the numerator: (2x^3 - 2x^2 + x + 1 + x + 1 - 2x^2) / (2x-2)(x+1)(x^2-1).

This simplifies to: (2x^3 - 4x^2 + 2x + 2) / (2x-2)(x+1)(x^2-1).