You need to bring the terms to the right, to a common denominator, such that:

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

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

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

Equating the coefficients of like powers yields:

`{(a + b = 3),(2a - 2b = 1):} => {(a = 3 - b),(2(3 - b) - 2b = 1):}`

`{(a = 3 - b),(6 - 2b - 2b = 1):} => {(a = 3 - b),(-4b = -6 + 1):}`

`b = 5/4`

Replacing `5/4` for b in `a = 3 - b` yields:

`a = 3 - 5/4 => a = 7/4`

**Hence, evaluating a and b, using partial fraction decomposition, yields **`a = 7/4, b = 5/4.`

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