\(\sqrt{x+3}+3x.\sqrt{x+5}=3x+\sqrt{x^2+8x+15}\)      (\(x\ge-3\))

\(\Leftrightarrow\sqrt{x+3}+3x.\sqrt{x+5}=3x+\sqrt{x+3}.\sqrt{x+5}\)

\(\Leftrightarrow\sqrt{x+3}.\left(1-\sqrt{x+5}\right)-3x.\left(1-\sqrt{x+5}\right)=0\)

\(\left(\sqrt{x+3}-3x\right).\left(1-\sqrt{x+5}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x+3=9x^2\\x+5=1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}9x^2-x-3=0\\x=-4\left(KTM\right)\end{matrix}\right.\)

\(\Leftrightarrow\left(3x\right)^2-2.3x.\dfrac{1}{6}+\dfrac{1}{36}-\dfrac{1}{36}-3=0\)

\(\Leftrightarrow\left(3x-\dfrac{1}{6}\right)^2-\dfrac{109}{36}=0\)

\(\Leftrightarrow\left(3x-\dfrac{1}{6}-\dfrac{\sqrt{109}}{6}\right).\left(3x-\dfrac{1}{6}+\dfrac{\sqrt{109}}{6}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1+\sqrt{109}}{18}\left(TM\right)\\x=\dfrac{1-\sqrt{109}}{18}\left(TM\right)\end{matrix}\right.\)

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