If `(3x^(2)+5)/((x^(2)+1)^2)= (a)/(x^(2)+1)+(b)/((x^(2)+1)^2)` then `(a, b)=`

Options

A. `(2,3)`
B. `(3,2)`
C. `(-2,3)`
D. `(-3,2)`

Answer

To solve the equation

\[ \frac{3x^2 + 5}{(x^2 + 1)^2} = \frac{a}{x^2 + 1} + \frac{b}{(x^2 + 1)^2} \]

we will express the right-hand side with a common denominator and then compare coefficients.

### Step 1: Rewrite the right-hand side with a common denominator

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

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

### Step 2: Set the numerators equal

Now we have:

\[ \frac{3x^2 + 5}{(x^2 + 1)^2} = \frac{a(x^2 + 1) + b}{(x^2 + 1)^2} \]

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

\[ 3x^2 + 5 = a(x^2 + 1) + b \]

### Step 3: Expand the right-hand side

Expanding the right-hand side gives:

\[ 3x^2 + 5 = ax^2 + a + b \]

### Step 4: Collect like terms

Now, we can rearrange this equation:

\[ 3x^2 + 5 = ax^2 + (a + b) \]

### Step 5: Compare coefficients

Now we can compare the coefficients of \(x^2\) and the constant terms on both sides:

1. Coefficient of \(x^2\):
\[ a = 3 \]

2. Constant term:
\[ a + b = 5 \]

### Step 6: Substitute to find \(b\)

Now that we know \(a = 3\), we can substitute this into the second equation:

\[ 3 + b = 5 \]

Subtracting 3 from both sides gives:

\[ b = 2 \]

### Final Result

Thus, the values of \(a\) and \(b\) are:

\[ (a, b) = (3, 2) \]