Step 1: Understanding the Concept:
This problem involves the decomposition of a rational expression into partial fractions. The goal is to find the sum of the numerators A and B of the decomposed fractions.
Step 2: Key Formula or Approach:
To find the values of A and B, we first combine the fractions on the right side over a common denominator:
\[ \frac{2x+5}{(x-1)(x+3)} = \frac{A(x+3) + B(x-1)}{(x-1)(x+3)} \]
By equating the numerators, we get the identity:
\[ 2x+5 = A(x+3) + B(x-1) \]
We can find A and B by substituting strategic values for x (the "cover-up" method) or by comparing coefficients.
Step 3: Detailed Explanation (Method 1: Cover-up Method):
To find A, we choose a value of x that makes the B term zero, which is $x=1$:
\[ 2(1) + 5 = A(1+3) + B(1-1) \]
\[ 7 = A(4) + B(0) \]
\[ 7 = 4A \implies A = \frac{7}{4} \]
To find B, we choose a value of x that makes the A term zero, which is $x=-3$:
\[ 2(-3) + 5 = A(-3+3) + B(-3-1) \]
\[ -6 + 5 = A(0) + B(-4) \]
\[ -1 = -4B \implies B = \frac{1}{4} \]
Now, calculate the required sum A + B:
\[ A + B = \frac{7}{4} + \frac{1}{4} = \frac{8}{4} = 2 \]
Step 3: Detailed Explanation (Method 2: Comparing Coefficients):
Expand the identity from Step 2:
\[ 2x+5 = Ax + 3A + Bx - B \]
Group the terms with x and the constant terms:
\[ 2x+5 = (A+B)x + (3A-B) \]
Now, compare the coefficients of the powers of x on both sides.
Comparing coefficients of x:
\[ A+B = 2 \]
Comparing constant terms:
\[ 3A-B = 5 \]
From the first equation, we directly get the answer A+B = 2.
Step 4: Final Answer:
Both methods yield A+B = 2. Therefore, option (B) is the correct answer.