To resolve the expression \(\frac{3x^2 + 1}{(x^2 - 3x + 2)(2x + 1)}\) into partial fractions, we will follow these steps:
### Step 1: Factor the Denominator
The denominator can be factored as follows:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]
Thus, the expression becomes:
\[
\frac{3x^2 + 1}{(x - 1)(x - 2)(2x + 1)}
\]
### Step 2: Set Up the Partial Fraction Decomposition
We express the fraction as a sum of partial fractions:
\[
\frac{3x^2 + 1}{(x - 1)(x - 2)(2x + 1)} = \frac{Ax + B}{x - 1} + \frac{C}{x - 2} + \frac{D}{2x + 1}
\]
Here, \(A\), \(B\), \(C\), and \(D\) are constants that we need to determine.
### Step 3: Combine the Right Side
To combine the right side, we need a common denominator:
\[
\frac{Ax + B}{x - 1} + \frac{C}{x - 2} + \frac{D}{2x + 1} = \frac{(Ax + B)(x - 2)(2x + 1) + C(x - 1)(2x + 1) + D(x - 1)(x - 2)}{(x - 1)(x - 2)(2x + 1)}
\]
### Step 4: Expand the Numerator
Now we expand the numerator:
\[
(Ax + B)(x - 2)(2x + 1) + C(x - 1)(2x + 1) + D(x - 1)(x - 2)
\]
This will yield a polynomial in \(x\).
### Step 5: Set Up the Equation
Now, we equate the numerators:
\[
3x^2 + 1 = (Ax + B)(x - 2)(2x + 1) + C(x - 1)(2x + 1) + D(x - 1)(x - 2)
\]
### Step 6: Collect Like Terms
Combine all terms on the right-hand side and collect like terms for \(x^2\), \(x\), and the constant term.
### Step 7: Create a System of Equations
By comparing coefficients of \(x^2\), \(x\), and the constant term from both sides, we can create a system of equations:
1. Coefficient of \(x^2\): \(2A + C + D = 3\)
2. Coefficient of \(x\): \(-2A + C - D = 0\)
3. Constant term: \(-2B + C = 1\)
### Step 8: Solve the System of Equations
Now we solve the system of equations for \(A\), \(B\), \(C\), and \(D\).
From equation 3:
\[
C = 1 + 2B
\]
Substituting \(C\) into equations 1 and 2 gives us two equations in terms of \(A\) and \(B\):
1. \(2A + (1 + 2B) + D = 3\)
2. \(-2A + (1 + 2B) - D = 0\)
### Step 9: Solve for \(A\), \(B\), \(C\), and \(D\)
After substituting and simplifying, we can find the values of \(A\), \(B\), \(C\), and \(D\).
### Step 10: Write the Final Partial Fraction Decomposition
Once we have the values of \(A\), \(B\), \(C\), and \(D\), we can substitute them back into the partial fractions:
\[
\frac{3x^2 + 1}{(x - 1)(x - 2)(2x + 1)} = \frac{Ax + B}{x - 1} + \frac{C}{x - 2} + \frac{D}{2x + 1}
\]
