将 $\frac{3x^2 + 5x - 2}{(x + 1)(x^2 + 4)}$ 化为部分分式。

1. Set up the correct form for the partial fractions

Let $\frac{3x^2 + 5x - 2}{(x + 1)(x^2 + 4)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 4}$

2. Form the basic algebraic identity

Multiply both sides by $(x + 1)(x^2 + 4)$:
$3x^2 + 5x - 2 = A(x^2 + 4) + (Bx + C)(x + 1)$

3. Find the value of $A$ by substitution

Substitute $x = -1$:
$3(-1)^2 + 5(-1) - 2 = A((-1)^2 + 4) + 0$
$3 - 5 - 2 = 5A$
$-4 = 5A \implies A = -\frac{4}{5}$

4. Find the value of $C$ by substitution

Substitute $x = 0$:
$-2 = 4A + C$
$-2 = 4\left(-\frac{4}{5}\right) + C$
$-2 = -\frac{16}{5} + C \implies C = \frac{6}{5}$

5. Find the value of $B$ by comparing coefficients

Compare coefficients of $x^2$:
$3 = A + B$
$3 = -\frac{4}{5} + B \implies B = \frac{19}{5}$