<s>[INST] Simplifying third degree polynomials I am trying to figure out this simple problem but I have no idea how to do it.
$\dfrac {2x^3 - 5x^2 -4x -3}{2x^3 + x^2 - 18x -9}$
I know that I can keep guessing at random to find a factor in the terms to factor them out but that seems like it would be far too time intensive, especially considering that if this was on a test it would likely be the first step of the problem and I can't spend 30 minutes just setting the problem up.
 [/INST] Any common divisor of $2x^3 - 5x^2 -4x -3$ and $2x^3 + x^2 - 18x -9$ should also be a common divisor of their difference.
$$(2x^3 + x^2 - 18x -9) - (2x^3 - 5x^2 -4x -3) 
  = 6x^2 - 14x - 6 = 2(3x^2 - 7x -3)$$
$3x^2 - 7x -3$ does not have any rational-coefficient factors and it does not divide evenly into $2x^3 - 5x^2 -4x -3$. That is enough information to conclude that 
$$\gcd(2x^3 - 5x^2 -4x -3, 2x^3 + x^2 - 18x -9) = 1$$
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