Resolve `(5x+6)/((2+x)(1-x))` into partial fractions.

Answer

To resolve the expression \(\frac{5x + 6}{(2 + x)(1 - x)}\) into partial fractions, we will follow these steps:

### Step 1: Set up the partial fraction decomposition
We start by expressing the given fraction as a sum of simpler fractions:
\[ \frac{5x + 6}{(2 + x)(1 - x)} = \frac{A}{2 + x} + \frac{B}{1 - x} \]
where \(A\) and \(B\) are constants that we need to determine.

### Step 2: Combine the right-hand side
To combine the right-hand side into a single fraction, we find a common denominator:
\[ \frac{A}{2 + x} + \frac{B}{1 - x} = \frac{A(1 - x) + B(2 + x)}{(2 + x)(1 - x)} \]
This gives us:
\[ \frac{A(1 - x) + B(2 + x)}{(2 + x)(1 - x)} = \frac{5x + 6}{(2 + x)(1 - x)} \]

### Step 3: Set the numerators equal
Since the denominators are the same, we can equate the numerators:
\[ A(1 - x) + B(2 + x) = 5x + 6 \]

### Step 4: Expand the left-hand side
Expanding the left-hand side, we get:
\[ A - Ax + 2B + Bx = 5x + 6 \]
Combining like terms results in:
\[ (-A + B)x + (A + 2B) = 5x + 6 \]

### Step 5: Set up a system of equations
Now, we can set up a system of equations by comparing coefficients:
1. For the coefficient of \(x\): \(-A + B = 5\)
2. For the constant term: \(A + 2B = 6\)

### Step 6: Solve the system of equations
We can solve these equations simultaneously. From the first equation, we can express \(B\) in terms of \(A\):
\[ B = A + 5 \]
Substituting \(B\) into the second equation:
\[ A + 2(A + 5) = 6 \]
This simplifies to:
\[ A + 2A + 10 = 6 \implies 3A + 10 = 6 \implies 3A = -4 \implies A = -\frac{4}{3} \]
Now substituting back to find \(B\):
\[ B = -\frac{4}{3} + 5 = -\frac{4}{3} + \frac{15}{3} = \frac{11}{3} \]

### Step 7: Write the partial fraction decomposition
Now that we have \(A\) and \(B\), we can write the partial fraction decomposition:
\[ \frac{5x + 6}{(2 + x)(1 - x)} = \frac{-\frac{4}{3}}{2 + x} + \frac{\frac{11}{3}}{1 - x} \]

### Final Answer
Thus, the resolved partial fractions are:
\[ \frac{5x + 6}{(2 + x)(1 - x)} = -\frac{4}{3(2 + x)} + \frac{11}{3(1 - x)} \]