\(\hept{\begin{cases}a+b+c=4\\a^2+b^2+c^2=6\end{cases}}\)
\(b^2+c^2=6-a^2\Rightarrow\left(b+c\right)^2-2bc=6-a^2\)
\(\Rightarrow2bc=\frac{\left(b+c\right)^2-6+a^2}{2}\)
\(=\frac{\left(4-a\right)^2-6+a^2}{2}\left(Do:a+b+c=4\right)\)
\(=\frac{2a^2-8a+10}{2}=a^2-4a+5\)
\(\Rightarrow P=a^3+bc\left(b+c\right)=a^3+\left(a^2-4a+5\right)\left(4-a\right)\left(Do:a+b+c=4\right)\)
\(=a^3+4a^2-16a+20-a^3+4a^2-5a\)
\(=8a^2-21a+20\)
\(=8\left(a^2-2.\frac{21}{16}a+\frac{441}{256}\right)+\frac{199}{32}\)
\(=8\left(a-\frac{21}{16}\right)^2+\frac{119}{32}\)
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