<s>[INST] How to factorize $\frac{4x^3+4x^2-7x+2}{4x^4-17x^2+4}$? How to factorize and simplify the following?
$$\frac{4x^3+4x^2-7x+2}{4x^4-17x^2+4}$$
I've tried everything I know.  Trying to factorize the numerator first then denominator, but I get no where.  Usual identities like $(x+y)^2=x^2+2xy+y^2$ don't work either, and neither does long division. I'm pretty stuck.
The answer from wolfram is 
$(2x-1)/((2x+1)(x-2))$.
But I can't get there.
 [/INST] Just above the line that says GCD it shows the quotients, which you would want because of wishing to reduce the fraction
$$  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  $$ 
$$  \left(  4 x^{4}  - 17 x^{2}  + 4 \right)  $$ 
$$  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  =  \left(  4 x^{4}  - 17 x^{2}  + 4 \right)  \cdot \color{magenta}{  \left( 0 \right) } +  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  $$
$$  \left(  4 x^{4}  - 17 x^{2}  + 4 \right)  =  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  \cdot \color{magenta}{  \left(   x  - 1 \right) } +  \left(   - 6 x^{2}  - 9 x  + 6 \right)  $$
$$  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  =  \left(   - 6 x^{2}  - 9 x  + 6 \right)  \cdot \color{magenta}{  \left(   \frac{  - 2 x  + 1 }{ 3 }  \right) } +  \left( 0 \right)  $$
$$ \frac{ 0}{1} $$
$$ \frac{ 1}{0} $$
$$ \color{magenta}{  \left( 0 \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left( 0 \right) }{ \left( 1  \right) } $$
$$ \color{magenta}{  \left(   x  - 1 \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left( 1  \right) }{ \left(   x  - 1 \right) } $$
$$ \color{magenta}{  \left(   \frac{  - 2 x  + 1 }{ 3 }  \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   \frac{  - 2 x  + 1 }{ 3 }  \right) }{ \left(   \frac{  - 2 x^{2}  + 3 x  + 2 }{ 3 }  \right) } $$
$$  \left(  2 x  - 1 \right)  \left(   \frac{  x  - 1 }{ 3 }  \right)  -  \left(  2 x^{2}  - 3 x  - 2 \right)  \left( \frac{ 1}{3 } \right)  =  \left( 1  \right)  $$
$$  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  =  \left(  2 x  - 1 \right)  \cdot \color{magenta}{  \left(  2 x^{2}  + 3 x  - 2 \right) } +  \left( 0 \right)  $$
$$  \left(  4 x^{4}  - 17 x^{2}  + 4 \right)  =  \left(  2 x^{2}  - 3 x  - 2 \right)  \cdot \color{magenta}{  \left(  2 x^{2}  + 3 x  - 2 \right) } +  \left( 0 \right)  $$
$$  \mbox{GCD} =   \color{magenta}{  \left(  2 x^{2}  + 3 x  - 2 \right) }   $$
$$  \left(  4 x^{3}  + 4 x^{2}  - 7 x  + 2 \right)  \left(   \frac{  x  - 1 }{ 3 }  \right)  -  \left(  4 x^{4}  - 17 x^{2}  + 4 \right)  \left( \frac{ 1}{3 } \right)  =  \left(  2 x^{2}  + 3 x  - 2 \right)  $$ 
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