Find the distance $d\left(P_{1}, P_{2}\right)$ between the given points $P_{1}$ and $P_{2}$.
\[
\begin{array}{l}
P_{1}=(4,2) \\
P_{2}=(-1,4)
\end{array}
\]

Step 1 ：We are given two points $P_{1}=(4,2)$ and $P_{2}=(-1,4)$ and we are asked to find the distance between these two points.

Step 2 ：The distance between two points in a 2D plane can be calculated using the distance formula derived from the Pythagorean theorem. The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 3 ：Substitute the given points into the formula: \[d = \sqrt{((-1) - 4)^2 + (4 - 2)^2}\]

Step 4 ：Simplify the expression inside the square root: \[d = \sqrt{(-5)^2 + 2^2}\]

Step 5 ：Calculate the squares: \[d = \sqrt{25 + 4}\]

Step 6 ：Add the numbers inside the square root: \[d = \sqrt{29}\]

Step 7 ：Calculate the square root: \[d = 5.385\]

Step 8 ：Final Answer: The distance between points $P_{1}$ and $P_{2}$ is \(\boxed{5.385}\)