To determine a and b, we'll have to calculate the least common denominator of the 3 ratios.

LCD = (x-2)(x+2) = x^2 - 4

LCD = x^2 - 4

Now, we'll multiply the first ratio from the right side by x+2 and the second ratio by x-2. The ratio from the left side has the denominator x^2 - 4, so it won't be multiplied.

We'll re-write the equation, all 3 quotients having the denominator x^2 - 4.

3x + 1 = a(x+2) + b(x-2)

We'll remove the brackets:

3x + 1 = ax + 2a + bx - 2b

We'll combine the terms from the right side with respect to x:

3x + 1 = x(a + b) + 2a - 2b

The expressions from both sides are equals if the correspondent coefficients are equal:

The coefficient of x from the left side has to be equal to the coefficient of x, from the right side:

a + b = 3 (1)

2a - 2b = 1 (2)

We'll multiply (1) by 2 and we'll get:

2a + 2b = 6 (3)

We'll add (3)+(2):

2a - 2b + 2a + 2b = 1 + 6

We'll combine and eliminate like terms:

4a = 7

**a = 7/4**

We'll substitute a in (1):

7/4 + b = 3

b = 3 - 7/4

b = (12-7)/4

**b = 5/4**

To find a and b in (3x+1)/(x^2-4)=a/(x-2)+b/(x+2).

We notice that the denominators on both sides have LCM (x-2)(x+2) = x^2-4.

So multiply both sides by (x-2)(x+2) and get:

3x+1 = a(x+2) +b(x-2).

3x+1 = ax+2a+bx-2b

3x = (a+b)x +2(a-b).

Now equate the like terms on both sides:

(a+b)x = 3x . Or a+b = 3.....(1).

2(a-b) = 1. Or a-b = 1/2......(2).

Eq(1) +Eq(2) gives: 2a = 3+1/2. So a = 7/2*2 = 7/4 .

Eq(1)- eq(2) gives: 2b = 3-1/2 = 5/2. Or b = 5/4.

Therefore a = 7/4 and b = 5/2.