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Here is an abstract algebra problem :

Solve the equation (in $ℚ[x]$) : $$f(x)(2x^3 + 3x^2 + 7x + 1) + g(x)(5x^4 + x + 1) = x + 3$$

From your comments, I tried to apply extended euclidean algo but it ends up with completely unlikely reminders and quotient. Can you please tell me what I'm doing wrong before I continue with these terrible calculations ?enter image description here

Thanks in advance.

AANICR
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  • Please edit your post to include how you found that, so that we can locate your mistake. – Anne Bauval Dec 17 '23 at 16:28
  • The standard method to solve such equation is by the extended Euclidean algorithm - see the linked dupe. Here the gcd $=1$ so it is solvable. – Bill Dubuque Dec 17 '23 at 17:20
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    @Nicrotte just as Bill said. The gcd is one. Use the extended to find the bezout-coeffiecents such that $a(x)(2x^3+3x^2+7x+1)+b(x)(5x^4+x+1)=1$, then multiply both sides by x+3 and you're done. – muhammed gunes Dec 17 '23 at 18:01

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