If `(3)/(( x+ 2) ( 2 x + 1) ) = (a)/( 2 x + 1) + (b)/( x + 2)` be an identify then the value of b is

Options

A. 0
B. `-1`
C. 2
D. 3

Answer

To solve the equation

\[ \frac{3}{(x + 2)(2x + 1)} = \frac{a}{2x + 1} + \frac{b}{x + 2} \]

we need to find the value of \( b \).

### Step 1: Find a common denominator for the right side

The common denominator for the right side is \((2x + 1)(x + 2)\). Therefore, we can rewrite the right side as:

\[ \frac{a(x + 2) + b(2x + 1)}{(2x + 1)(x + 2)} \]

### Step 2: Set the numerators equal

Now, we can equate the numerators of both sides:

\[ 3 = a(x + 2) + b(2x + 1) \]

### Step 3: Expand the right side

Expanding the right side gives:

\[ 3 = ax + 2a + 2bx + b \]

Combining like terms results in:

\[ 3 = (a + 2b)x + (2a + b) \]

### Step 4: Set up a system of equations

Since this equation must hold for all \( x \), we can equate the coefficients of \( x \) and the constant terms:

1. Coefficient of \( x \): \( a + 2b = 0 \)
2. Constant term: \( 2a + b = 3 \)

### Step 5: Solve the system of equations

From the first equation, we can express \( a \) in terms of \( b \):

\[ a = -2b \]

Now, substitute \( a = -2b \) into the second equation:

\[ 2(-2b) + b = 3 \]

This simplifies to:

\[ -4b + b = 3 \]

\[ -3b = 3 \]

Dividing both sides by -3 gives:

\[ b = -1 \]

### Conclusion

Thus, the value of \( b \) is:

\[ \boxed{-1} \]