How to decompose this expression as a sum of partial fractions: (3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) ?

To decompose the expression (3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) into partial fractions, we first need to factor the denominator.

The denominator x^4 + 4x^3 + 6x^2 + 5x + 2 can be factored as (x^2 + 2x + 1)(x^2 + 2x + 2).

Now we can rewrite the expression as:

(3x^2 + 2x - 5)/[(x^2 + 2x + 1)(x^2 + 2x + 2)]

Next, we need to express the fraction as a sum of partial fractions. We assume that the expression can be written as:

A/(x^2 + 2x + 1) + B/(x^2 + 2x + 2)

Now we need to find the values of A and B. To do this, we multiply both sides by the denominator and simplify:

(3x^2 + 2x - 5) = A(x^2 + 2x + 2) + B(x^2 + 2x + 1)

Expanding and simplifying, we get:

3x^2 + 2x - 5 = (A + B)x^2 + (2A + B)x + 2A

Now we can equate the coefficients of like terms:

A + B = 3
2A + B = 2
2A = -5

Solving these equations, we get A = -5/2 and B = 11/2.

Therefore, the expression (3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) can be decomposed into partial fractions as:

(-5/2)/(x^2 + 2x + 1) + (11/2)/(x^2 + 2x + 2)

Step 1: Factor the Denominator

```
x^4 + 4x^3 + 6x^2 + 5x + 2 = (x^2 + 2x + 1)(x^2 + 2x + 2)
```

Step 2: Write the Partial Fraction Decomposition

```
(3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) = A/(x^2 + 2x + 1) + B/(x^2 + 2x + 2)
```

where A and B are constants that we will solve for.

Step 3: Clear the Denominators

```
3x^2 + 2x - 5 = A(x^2 + 2x + 2) + B(x^2 + 2x + 1)
```

Step 4: Equate Coefficients of Like Powers

- x^2 Term: 3 = A + B
- x Term: 2 = 2A + 2B
- Constant Term: -5 = 2A + B

Step 5: Solve for A and B

Solving the system of equations gives:

- A = -2
- B = 5

Step 6: Substitute A and B

```
(3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) = -2/(x^2 + 2x + 1) + 5/(x^2 + 2x + 2)
```

Therefore, the partial fraction decomposition of the given expression is:

```
(3x^2 + 2x - 5)/(x^4 + 4x^3 + 6x^2 + 5x + 2) = -2/(x^2 + 2x + 1) + 5/(x^2 + 2x + 2)
```