The partial fraction decomposition of $\frac{14 x}{8 x^{2}-10 x+3}$ can be written in the form of $\frac{f(x)}{2 x-1}+\frac{g(x)}{4 x-3}$, where
\[
\begin{array}{l}
f(x)= \\
g(x)=
\end{array}
\]

Step 1 ：The partial fraction decomposition of \(\frac{14 x}{8 x^{2}-10 x+3}\) can be written in the form of \(\frac{f(x)}{2 x-1}+\frac{g(x)}{4 x-3}\), where \(f(x)\) and \(g(x)\) are functions to be determined.

Step 2 ：To find the functions \(f(x)\) and \(g(x)\), we can equate the given fraction with the sum of the two simpler fractions and solve for \(f(x)\) and \(g(x)\).

Step 3 ：We can write the equation as follows: \(\frac{14x}{8x^2 - 10x + 3} = \frac{f(x)}{2x - 1} + \frac{g(x)}{4x - 3}\)

Step 4 ：We can multiply both sides by \(8x^2 - 10x + 3\) to clear the fraction: \(14x = f(x)(4x - 3) + g(x)(2x - 1)\)

Step 5 ：This is a system of linear equations in terms of \(f(x)\) and \(g(x)\). We can solve this system to find the values of \(f(x)\) and \(g(x)\).

Step 6 ：The solution to the equation is \(f: -7, g: 21\). This means that the functions \(f(x)\) and \(g(x)\) are constants, with \(f(x) = -7\) and \(g(x) = 21\).

Step 7 ：Therefore, the partial fraction decomposition of the given fraction is \(\frac{-7}{2x - 1} + \frac{21}{4x - 3}\).

Step 8 ：Final Answer: \[\begin{array}{l} f(x) = \boxed{-7} \\ g(x) = \boxed{21} \end{array}\]