Find polynomials $f(x),g(x)$ such that $f(x)p(x) + g(x)q(x) = 2x^2 + 6x +2$ Let $$p(x) = x^4 + 7x^3 + 14x^2 + 7x+1 $$$$q(x) = x^4 +10x^3 + 23x^2 + 10x+1$$ 
Find polynomials $f(x),g(x)$ with rational coefficients such that $$f(x)p(x) + g(x)q(x) = 2x^2 + 6x +2$$
I totally have no idea to solve this problem... Please help me to do it. Thank you!!

You can use Grobner basis to solve this,
$f1=x^4+7x^3+14x^2+7x+1;$,
$f2=x^4+10x^3+23x^2+10x+1;$
$f3=(f2-f1)/3=x^2+3x^2+x$
$f4=-3(f2-(x+7)f3)=x^2+3x+1$
Since,
$2x^2+6x+2 = 2*f4$
Now just replace it as $f4->f3->f2$ till everything is in terms of $f1$ and  $f2$,
Answer:  $2/3(-(x+4)f2+(x+7)f1)$ 
