If `(5x+6)/((2+x)(1-x))=a/(2+x)+b/(1-x)`, then the values of a and b respectively are:

Options

A. `-5/3,6/5`
B. `5/3,-6/5`
C. `-4/3,11/3`
D. `4/3,11/3`

Answer

To solve the equation \(\frac{5x + 6}{(2 + x)(1 - x)} = \frac{a}{2 + x} + \frac{b}{1 - x}\), we will follow these steps:

### Step 1: Find a common denominator for the right-hand side
The common denominator for the right-hand side is \((2 + x)(1 - x)\). We can rewrite the right-hand side as:
\[ \frac{a(1 - x) + b(2 + x)}{(2 + x)(1 - x)} \]

### Step 2: Set the numerators equal
Since the denominators are equal, we can set the numerators equal to each other:
\[ 5x + 6 = a(1 - x) + b(2 + x) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives:
\[ 5x + 6 = a - ax + 2b + bx \]
Combining like terms results in:
\[ 5x + 6 = (b - a)x + (a + 2b) \]

### Step 4: Create a system of equations
Now we can create a system of equations by comparing coefficients from both sides:
1. Coefficient of \(x\): \(b - a = 5\) (Equation 1)
2. Constant term: \(a + 2b = 6\) (Equation 2)

### Step 5: Solve the system of equations
From Equation 1, we can express \(a\) in terms of \(b\):
\[ a = b - 5 \]

Now substitute this expression for \(a\) into Equation 2:
\[ (b - 5) + 2b = 6 \]
Combining like terms gives:
\[ 3b - 5 = 6 \]
Adding 5 to both sides results in:
\[ 3b = 11 \]
Dividing by 3 gives:
\[ b = \frac{11}{3} \]

### Step 6: Substitute back to find \(a\)
Now substitute \(b = \frac{11}{3}\) back into the expression for \(a\):
\[ a = \frac{11}{3} - 5 = \frac{11}{3} - \frac{15}{3} = -\frac{4}{3} \]

### Final Answer
Thus, the values of \(a\) and \(b\) are:
\[ a = -\frac{4}{3}, \quad b = \frac{11}{3} \]