Resolve `(2x^(4)+2x^(2)+x+1)/(x(x^(2)+1)^(2))` into partial fractions.

Answer

To resolve the expression \(\frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2}\) into partial fractions, we will follow these steps:

### Step 1: Set up the partial fraction decomposition
We start by expressing the given fraction in terms of its partial fractions. The denominator \(x(x^2 + 1)^2\) suggests the following form:

\[ \frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2} \]

### Step 2: Multiply through by the common denominator
To eliminate the denominators, multiply both sides by \(x(x^2 + 1)^2\):

\[ 2x^4 + 2x^2 + x + 1 = A(x^2 + 1)^2 + (Bx + C)x(x^2 + 1) + (Dx + E)x \]

### Step 3: Expand the right-hand side
Now, we expand the right-hand side:

1. **Expand \(A(x^2 + 1)^2\)**:
\[ A(x^2 + 1)^2 = A(x^4 + 2x^2 + 1) \]

2. **Expand \((Bx + C)x(x^2 + 1)\)**:
\[ (Bx + C)x(x^2 + 1) = (Bx + C)(x^3 + x) = Bx^4 + Cx^3 + Bx^2 + Cx \]

3. **Expand \((Dx + E)x\)**:
\[ (Dx + E)x = Dx^2 + Ex \]

Combining all these expansions gives:

\[ 2x^4 + 2x^2 + x + 1 = Ax^4 + (2A + C + B)x^2 + (B + D)x^3 + (A + E) \]

### Step 4: Collect like terms
Now we can collect like terms from the right-hand side:

- Coefficient of \(x^4\): \(A + B = 2\)
- Coefficient of \(x^3\): \(C = 0\)
- Coefficient of \(x^2\): \(2A + B + D = 2\)
- Coefficient of \(x\): \(B + E = 1\)
- Constant term: \(A = 1\)

### Step 5: Solve the system of equations
From the constant term, we have:
\[ A = 1 \]
Substituting \(A = 1\) into the first equation:
\[ 1 + B = 2 \implies B = 1 \]
Substituting \(B = 1\) into the equation for \(x^2\):
\[ 2(1) + 1 + D = 2 \implies 2 + D = 2 \implies D = 0 \]
Since \(C = 0\) from the \(x^3\) coefficient, we can substitute \(B = 1\) into the equation for \(x\):
\[ 1 + E = 1 \implies E = 0 \]

### Step 6: Write the final partial fraction decomposition
Now we have:
- \(A = 1\)
- \(B = 1\)
- \(C = 0\)
- \(D = 0\)
- \(E = 0\)

Thus, the partial fraction decomposition is:
\[ \frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2} = \frac{1}{x} + \frac{x}{x^2 + 1} \]

### Final Answer
\[ \frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2} = \frac{1}{x} + \frac{x}{x^2 + 1} \]