If `(3)/((x+2)(2x+1))=(a)/(2x+1)+(b)/(x+2)` then b= ?

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
The common denominator for the right-hand side is \( (2x + 1)(x + 2) \).

### Step 2: Rewrite the right-hand side
We can rewrite the right-hand side as:

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

### Step 3: Set the numerators equal
Since the denominators are the same, we can set the numerators equal to each other:

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

### Step 4: Expand the right-hand side
Expanding the right-hand side gives:

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

### Step 5: Combine like terms
Combining like terms results in:

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

### Step 6: Compare coefficients
Now we can compare coefficients from both sides:

1. The coefficient of \( x \): \( a + 2b = 0 \) (Equation 1)
2. The constant term: \( 2a + b = 3 \) (Equation 2)

### Step 7: Solve the system of equations
From Equation 1, we can express \( a \) in terms of \( b \):

\[ a = -2b \]

Now substitute \( a \) into Equation 2:

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

This simplifies to:

\[ -4b + b = 3 \]
\[ -3b = 3 \]
\[ b = -1 \]

### Final Answer
Thus, the value of \( b \) is

\[ \boxed{-1} \]

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