Resolve into partial fractions :
`(2x^(4)+2x^(2)+x+1)/(x(x^(2)+1)^2)=(A)/(x)+(Bx+C)/((x^(2)+1)^2)+(Ex+F)/((x^(2)+1))`

Answer

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

### Step 1: Set up the Partial Fraction Decomposition
We start by expressing the given fraction in terms of its partial fractions:
\[ \frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2} = \frac{A}{x} + \frac{Bx + C}{(x^2 + 1)^2} + \frac{Ex + F}{x^2 + 1} \]

### Step 2: Find a Common Denominator
The common denominator for the right-hand side is \(x(x^2 + 1)^2\). We can rewrite the right-hand side as:
\[ \frac{A(x^2 + 1)^2 + (Bx + C)x + (Ex + F)x(x^2 + 1)}{x(x^2 + 1)^2} \]

### Step 3: Expand the Numerator
Now we need to expand the numerator:
1. Expand \(A(x^2 + 1)^2\):
\[ A(x^4 + 2x^2 + 1) = Ax^4 + 2Ax^2 + A \]
2. Expand \((Bx + C)x\):
\[ Bx^2 + Cx \]
3. Expand \((Ex + F)x(x^2 + 1)\):
\[ (Ex + F)(x^3 + x) = Ex^4 + Fx^3 + Ex^2 + Fx \]

Combining these, the numerator becomes:
\[ Ax^4 + 2Ax^2 + A + Bx^2 + Cx + Ex^4 + Fx^3 + Ex^2 + Fx \]

### Step 4: Combine Like Terms
Now, combine the coefficients of like terms:
\[ (A + E)x^4 + (2A + B + E)x^2 + (C + F)x + A + 0 \]

### Step 5: Set Up the Equation
We equate this to the numerator of the left-hand side:
\[ 2x^4 + 0x^3 + 2x^2 + x + 1 \]
This gives us the following system of equations:
1. \(A + E = 2\) (coefficient of \(x^4\))
2. \(F = 0\) (coefficient of \(x^3\))
3. \(2A + B + E = 2\) (coefficient of \(x^2\))
4. \(C + F = 1\) (coefficient of \(x\))
5. \(A = 1\) (constant term)

### Step 6: Solve the System of Equations
From equation 5, we have:
\[ A = 1 \]
Substituting \(A = 1\) into equation 1:
\[ 1 + E = 2 \implies E = 1 \]
Substituting \(A = 1\) and \(E = 1\) into equation 3:
\[ 2(1) + B + 1 = 2 \implies B = -1 \]
From equation 4, since \(F = 0\):
\[ C + 0 = 1 \implies C = 1 \]

### Step 7: Write the Final Partial Fraction Decomposition
Now we can substitute the values back into the partial fractions:
\[ \frac{2x^4 + 2x^2 + x + 1}{x(x^2 + 1)^2} = \frac{1}{x} + \frac{-x + 1}{(x^2 + 1)^2} + \frac{x}{x^2 + 1} \]

### Final Result
Thus, the partial fraction decomposition is:
\[ \frac{1}{x} + \frac{-x + 1}{(x^2 + 1)^2} + \frac{x}{x^2 + 1} \]