Ta có: \(\left|2x+3y\right|\ge0\)\(\forall x,y\inℝ\); \(\left|4y+5z\right|\ge0\)\(\forall y,z\inℝ\); \(\left|xy+yz+zx+110\right|\ge0\)\(\forall x,y,z\inℝ\)

Nên: \(P=\left|2x+3y\right|+\left|4y+5z\right|+\left|xy+yz+xz+110\right|\ge0\)\(\forall x,y,z\inℝ\)

Dấu " = " xảy ra <=> \(\left|2x+3y\right|+\left|4y+5z\right|+\left|xy+yz+xz+110\right|=0\)

Có: \( \left|2x+3y\right|=0\)\(\Leftrightarrow2x+3y=0\)\(\Leftrightarrow2x=-3y\)\(\Leftrightarrow\frac{x}{-3}=\frac{y}{2}\)

\(\left|4y+5z\right|=0\)\(\Leftrightarrow4y+5z=0\)\(\Leftrightarrow4y=-5z\)\(\Leftrightarrow\frac{y}{-5}=\frac{z}{4}\)

\(\left|xy+yz+zx+110\right|=0\)\(\Leftrightarrow xy+yz+zx+110=0\)\(\Leftrightarrow xy+yz+zx=-110\)

Lại có: \(\frac{x}{-3}=\frac{y}{2}\)\(\Rightarrow\frac{x}{15}=\frac{y}{-10}\) (1) ;  \(\frac{y}{-5}=\frac{z}{4}\)\(\Rightarrow\frac{y}{-10}=\frac{z}{8}\)(2)

Từ (1) và (2) \(\Rightarrow\frac{x}{15}=\frac{y}{-10}=\frac{z}{8}=k\)=> x = 15k ; y = (-10) . k ; z = 8k

Ta có: \(xy+yz+zx=-110\)\(\Rightarrow15k\left(-10\right)k+8k\left(-10\right)k+8k.15k=-110\)

\(\Rightarrow k^2\left(-150\right)+k^2\left(-80\right)+120k^2=-110\)

\(\Rightarrow k^2\left(-110\right)=-110\)\(\Rightarrow k^2=1\)\(\Rightarrow\orbr{\begin{cases}k=1\\k=-1\end{cases}}\)

+) Th1: k = 1   

Có: x = 15k = 15 . 1 = 15

y = (-10) . k = (-10) . 1 = -10

z = 8k = 8 . 1 = 8

+) Th2: k = -1

Có: x = 15k = 15 . (-1) = -15 

y = (-10) . k = (-10) . (-1) = 10

z = 8k = 8 . (-1) = -8

Vậy GTNN P = 0 <=> (x; y; z) = (15; -10; 8) hoặc (x; y; z) = (-15; 10; -8)

k bít