If `(ax+b)/((3x+4)^2)=(1)/((3x+4))-(3)/((3x+4)^2)` then

Options

A. `a+b=4`
B. `a-b=3`
C. `a-b=2`
D. `a+b=5`

Answer

To solve the equation

\[ \frac{ax+b}{(3x+4)^2} = \frac{1}{3x+4} - \frac{3}{(3x+4)^2} \]

we will follow these steps:

### Step 1: Rewrite the Right-Hand Side
First, we need to combine the right-hand side into a single fraction. The right-hand side is

\[ \frac{1}{3x+4} - \frac{3}{(3x+4)^2} \]

To combine these fractions, we need a common denominator, which is \((3x+4)^2\):

\[ \frac{1 \cdot (3x+4) - 3}{(3x+4)^2} = \frac{3x + 4 - 3}{(3x+4)^2} = \frac{3x + 1}{(3x+4)^2} \]

### Step 2: Set the Numerators Equal
Now we have:

\[ \frac{ax+b}{(3x+4)^2} = \frac{3x + 1}{(3x+4)^2} \]

Since the denominators are the same, we can set the numerators equal to each other:

\[ ax + b = 3x + 1 \]

### Step 3: Compare Coefficients
Next, we compare the coefficients of \(x\) and the constant terms on both sides:

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

### Step 4: Conclusion
Thus, we find:

\[ a = 3 \quad \text{and} \quad b = 1 \]

### Step 5: Verify Conditions
We can check if these values satisfy the conditions given in the problem:

1. \(A + B = 3 + 1 = 4\) (True)
2. \(A - B = 3 - 1 = 2\) (True)

### Final Answer
The values of \(a\) and \(b\) are:

\[ a = 3, \quad b = 1 \]

---