To express \(\frac{7x^2 - 3x + 2}{x(x^2 + 1)}\) in partial fractions, we assume the form:

\(\frac{A}{x} + \frac{Bx + C}{x^2 + 1}\)

Multiply through by the denominator \(x(x^2 + 1)\) to clear the fractions:

\(7x^2 - 3x + 2 = A(x^2 + 1) + (Bx + C)x\)

Expanding the right side gives:

\(7x^2 - 3x + 2 = Ax^2 + A + Bx^2 + Cx\)

Combine like terms:

\(7x^2 - 3x + 2 = (A + B)x^2 + Cx + A\)

Equate coefficients:

1. \(A + B = 7\)

2. \(C = -3\)

3. \(A = 2\)

From equation 3, \(A = 2\).

Substitute \(A = 2\) into equation 1:

\(2 + B = 7\)

\(B = 5\)

Thus, the partial fraction decomposition is:

\(\frac{2}{x} + \frac{5x - 3}{x^2 + 1}\)