Use Cauchy:

P\(=\)\(\dfrac{4}{5}\left(x+y\right)+\left(\dfrac{6}{5}x+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\)\(\ge\dfrac{4}{5}.10+2\sqrt{\dfrac{6}{5}x.\dfrac{30}{x}}+2\sqrt{\dfrac{y}{5}.\dfrac{5}{y}}\)

\(=8+12+2=22\)

minP=22\(\Leftrightarrow x=y=5\)

 \(P=2x+y+\dfrac{30}{x}+\dfrac{5}{y}=\left(\dfrac{4}{5}x+\dfrac{6}{5}x\right)+\left(\dfrac{4}{5}y+\dfrac{1}{5}y\right)+\dfrac{30}{x}+\dfrac{5}{y}\)

\(=\left(\dfrac{4}{5}x+\dfrac{4}{5}y\right)+\left(\dfrac{6x}{5}+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\)

\(\dfrac{4}{5}\left(x+y\right)+\left(\dfrac{6x}{5}+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\)

Because x; y is positive we have :

\(\dfrac{6x}{5}+\dfrac{30}{x}\ge2\sqrt{\dfrac{6x}{5}.\dfrac{30}{x}}=2\sqrt{36}=12\) (Cauchy)

\(\dfrac{y}{5}+\dfrac{5}{y}\ge2\sqrt{\dfrac{y}{5}.\dfrac{5}{y}}=2\)(Caychy)

\(\Rightarrow\left(\dfrac{6x}{5}+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\ge14\)

We have \(x+y\ge10\Rightarrow\dfrac{4}{5}\left(x+y\right)\ge\dfrac{4}{5}.10=8\)

\(\Rightarrow\dfrac{4}{5}\left(x+y\right)+\left(\dfrac{6x}{5}+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)\ge14+8=22\)

The equation occurs when \(x=y=5\)

\(P_{min}=22\Leftrightarrow x=y=5\)