How to find out if a polynomial has multiple roots? Find out if polynomials have multiple roots:
1, $x^6-6x^4-4x^3+9x^2+12x+4 \in {\displaystyle \mathbb {Q}}$[$x$] 
2, $x^7+x^4+x^3+x+2\in {\displaystyle \mathbb {Z_3}}$[$x$]
I tried to differentiate these polynomials, but I haven't had any results. I know if a polynomial has double roots, then the first derivative of these polynomials has this root as a simple, etc. 
Thank you for any help!

just the first one. The second polynomial is the derivative of the first but divided through by $6.$ This is the extended Euclidean algorithm, where I write the back-substitution phase in the style of a continued fraction. 
$$  \left(   x^{6}  - 6 x^{4}  - 4 x^{3}  + 9 x^{2}  + 12 x  + 4 \right)  $$ 
$$  \left(   x^{5}  - 4 x^{3}  - 2 x^{2}  + 3 x  + 2 \right)  $$ 
$$  \left(   x^{6}  - 6 x^{4}  - 4 x^{3}  + 9 x^{2}  + 12 x  + 4 \right)  =  \left(   x^{5}  - 4 x^{3}  - 2 x^{2}  + 3 x  + 2 \right)  \cdot \color{magenta}{  \left(   x  \right) } +  \left(   - 2 x^{4}  - 2 x^{3}  + 6 x^{2}  + 10 x  + 4 \right)  $$ 
 $$  \left(   x^{5}  - 4 x^{3}  - 2 x^{2}  + 3 x  + 2 \right)  =  \left(   - 2 x^{4}  - 2 x^{3}  + 6 x^{2}  + 10 x  + 4 \right)  \cdot \color{magenta}{  \left(   \frac{  -  x  + 1 }{ 2 }  \right) } +  \left( 0 \right)  $$ 
 $$ \frac{ 0}{1} $$ 
 $$ \frac{ 1}{0} $$ 
 $$ \color{magenta}{  \left(   x  \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   x  \right) }{ \left( 1  \right) } $$ 
 $$ \color{magenta}{  \left(   \frac{  -  x  + 1 }{ 2 }  \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   \frac{  -  x^{2}  +  x  + 2 }{ 2 }  \right) }{ \left(   \frac{  -  x  + 1 }{ 2 }  \right) } $$ 
 $$  \left(   x^{2}  -  x  - 2 \right)  \left( \frac{ 1}{2 } \right)  -  \left(   x  - 1 \right)  \left(   \frac{  x  }{ 2 }  \right)  =  \left( -1  \right)  $$ 
 $$  \left(   x^{6}  - 6 x^{4}  - 4 x^{3}  + 9 x^{2}  + 12 x  + 4 \right)  =  \left(   x^{2}  -  x  - 2 \right)  \cdot \color{magenta}{  \left(   x^{4}  +  x^{3}  - 3 x^{2}  - 5 x  - 2 \right) } +  \left( 0 \right)  $$ 
 $$  \left(   x^{5}  - 4 x^{3}  - 2 x^{2}  + 3 x  + 2 \right)  =  \left(   x  - 1 \right)  \cdot \color{magenta}{  \left(   x^{4}  +  x^{3}  - 3 x^{2}  - 5 x  - 2 \right) } +  \left( 0 \right)  $$ 
 $$  \mbox{GCD} =   \color{magenta}{  \left(   x^{4}  +  x^{3}  - 3 x^{2}  - 5 x  - 2 \right) }   $$ 
 $$  \left(   x^{6}  - 6 x^{4}  - 4 x^{3}  + 9 x^{2}  + 12 x  + 4 \right)  \left( \frac{ 1}{2 } \right)  -  \left(   x^{5}  - 4 x^{3}  - 2 x^{2}  + 3 x  + 2 \right)  \left(   \frac{  x  }{ 2 }  \right)  =  \left(   -  x^{4}  -  x^{3}  + 3 x^{2}  + 5 x  + 2 \right)  $$ 
