`x^3+8y^3 +4x+8y`

`=(x+2y)(x^2 - 2xy + 4y^2 ) +4 (x+2y)`

`= (x+2y) (x^2 -2xy +4y^2 +4)`

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`x^2 +6xy +9y^2 +5x+15y`

`= (x+3y)^2 + 5(x+3y)`

`=(x+3y)( x+3y +5)`

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`25-x^2 +2xy -y^2`

`= 25 - (x^2 -2xy+y^2)`

`=25-(x-y)^2`

`=(5-x+y)(5+x-y)`

`x^3 -5x^2 -9x+45=0`

`<=> (x^3 -5x^2 )-(9x-45)=0`

`<=> x^2 (x-5)- 9(x-5)=0`

`<=>(x-5)(x^2 -9)=0`

`<=>(x-5)(x-3)(x+3)=0`

\(\Leftrightarrow\left[{}\begin{matrix}x-5=0\\x-3=0\\x+3=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=5\\x=3\\x=-3\end{matrix}\right.\)

\(3\sqrt{x-2}-\sqrt{4x-8}+4\sqrt{\dfrac{9x-18}{4}}=14\left(x\ge0;x\ne2\right)\\ \Leftrightarrow3\sqrt{x-2}-\sqrt{4\left(x-2\right)}+4\cdot\dfrac{1}{2}\sqrt{9\left(x-2\right)}=14\\ \Leftrightarrow3\sqrt{x-2}-2\sqrt{x-2}+6\sqrt{x-2}=14\\ \Leftrightarrow7\sqrt{x-2}=14\\ \Leftrightarrow\sqrt{x-2}=2\\ \Leftrightarrow x-2=4\\ \Leftrightarrow x=6\left(tm\right)\)