Express $\frac{15x^2 - 13x + 18}{(x - 2)(3x^2 + 1)}$ in partial fractions.

1. Set up the partial fraction decomposition form

Let $\frac{15x^2 - 13x + 18}{(x - 2)(3x^2 + 1)} = \frac{A}{x - 2} + \frac{Bx + C}{3x^2 + 1}$

2. Form the basic equation by clearing the denominator

$15x^2 - 13x + 18 = A(3x^2 + 1) + (Bx + C)(x - 2)$

3. Substitute $x = 2$ to find $A$

Let $x = 2$:
$15(2)^2 - 13(2) + 18 = A(3(2)^2 + 1) + 0$
$60 - 26 + 18 = 13A$
$52 = 13A \implies A = 4$

4. Substitute $x = 0$ to find $C$

Let $x = 0$:
$18 = A(1) + C(-2)$
$18 = 4 - 2C$
$2C = -14 \implies C = -7$

5. Compare coefficients of $x^2$ to find $B$

Compare coefficients of $x^2$:
$15 = 3A + B$
$15 = 3(4) + B$
$15 = 12 + B \implies B = 3$

6. Write the final partial fraction expression

$\frac{15x^2 - 13x + 18}{(x - 2)(3x^2 + 1)} = \frac{4}{x - 2} + \frac{3x - 7}{3x^2 + 1}$