Final answer:
The partial fraction decomposition of (3x+4)/((x+4)(4x-1)) involves finding constants A and B, so that the expression can be written as A/(x+4) + B/(4x-1).
Step-by-step explanation:
To find the partial fraction decomposition of the given function (3x+4)/((x+4)(4x-1)), we must express it as a sum of fractions whose denominators are the factors of the original denominator. The general form of the decomposition will be:
A/(x+4) + B/(4x-1)
We then find constants A and B such that:
(3x+4) = A(4x-1) + B(x+4)
This is achieved by solving a system of equations obtained by equating coefficients or by plugging in suitable x-values to solve for A and B.
After determining A and B, we rewrite the original function as the sum of the two fractions with the found coefficients.