The denominator is \( x(x^2 + x + 7) \), where \( x \) is a linear factor and \( x^2 + x + 7 \) is an irreducible quadratic. The partial fraction decomposition is:

\[ \frac{3x^2 + 7x + 28}{x(x^2 + x + 7)} = \frac{A}{x} + \frac{Bx + C}{x^2 + x + 7} \]

The least common denominator (LCD) is \( x(x^2 + x + 7) \). Combine the fractions:

\[ \frac{A(x^2 + x + 7) + (Bx + C)(x)}{x(x^2 + x + 7)} \]

Set the numerators equal:

\[ 3x^2 + 7x + 28 = A(x^2 + x + 7) + (Bx + C)(x) \]

Expand the right-hand side:

\[ A(x^2 + x + 7) + (Bx + C)(x) = Ax^2 + Ax + 7A + Bx^2 + Cx = (A + B)x^2 + (A + C)x + 7A \]

Equate coefficients of corresponding powers of \( x \):

\[ (A + B)x^2 + (A + C)x + 7A = 3x^2 + 7x + 28 \]

– Coefficient of \( x^2 \): \( A + B = 3 \)

– Coefficient of \( x \): \( A + C = 7 \)

– Constant: \( 7A = 28 \)

Solve the system of equations:

From \( 7A = 28 \), we get \( A = 4 \).

Substitute \( A = 4 \):

– \( 4 + B = 3 \Rightarrow B = -1 \)

– \( 4 + C = 7 \Rightarrow C = 3 \)

The partial fraction decomposition is:

\[ \frac{3x^2 + 7x + 28}{x(x^2 + x + 7)} = \frac{4}{x} + \frac{-x + 3}{x^2 + x + 7} \]

\[ \boxed{\frac{4}{x} + \frac{3 – x}{x^2 + x + 7}} \]