Final answer:
The partial fraction decomposition of the given rational function is done by expressing it as a sum of simpler fractions whose denominators are factors of the original denominator. The values in the decomposed form are found by equating numerators after finding a common denominator and solving for the unknown constants.
Step-by-step explanation:
The partial fraction decomposition of \( \frac{11x - 38}{(2x - 5)(x - 4)} \) involves breaking down the given rational expression into a sum of simpler fractions, where the denominators are factors of the original denominator. To find the correct partial fraction decomposition, one would typically write:
\( \frac{11x - 38}{(2x - 5)(x - 4)} = \frac{A}{2x - 5} + \frac{B}{x - 4} \)
Then, by finding a common denominator and equating the numerators, the values of A and B can be solved for. This involves equating coefficients for corresponding powers of x and solving the resulting system of equations. The correct decomposition would be a sum of fractions such that when combined, they give the original rational expression.