To express \(\frac{4x^2 - 13x + 13}{(2x - 1)(x - 3)}\) in partial fractions, we start by assuming:

\(\frac{4x^2 - 13x + 13}{(2x - 1)(x - 3)} = \frac{A}{2x - 1} + \frac{B}{x - 3}\)

Multiply through by \((2x - 1)(x - 3)\) to clear the denominators:

\(4x^2 - 13x + 13 = A(x - 3) + B(2x - 1)\)

Expanding both sides gives:

\(4x^2 - 13x + 13 = Ax - 3A + 2Bx - B\)

Combine like terms:

\(4x^2 - 13x + 13 = (A + 2B)x - (3A + B)\)

Equating coefficients, we get:

\(A + 2B = -13\)

\(-3A - B = 13\)

Solving these equations simultaneously, we find:

\(A = 2\), \(B = -3\)

Thus, the partial fraction decomposition is:

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

Alternatively, dividing the numerator by the denominator gives:

\(2 + \frac{-x + 7}{(2x - 1)(x - 3)}\)