Find Highest Common Factor of two polynomials $\ 2(x^4+9)-5x^2(x+1),\ 2x^3(2x-9)+81(x-1)$ Find the H.C.F. of $\ 2(x^4+9)-5x^2(x+1),\ 2x^3(2x-9)+81(x-1)$
I can't really see any pattern by inspection, so I carry out the multiplication with the brackets: $2x^4+18-5x^3-5x^2,\ 4x^4-18x^3+81x-81$
I tried to divide them into each other to find the common factor (Euler's method or something? I forgot). I did the obvious first step by multiplying $2x^4+18-5x^3-5x^2$ by $2$, then subtracting that from $4x^4-18x^3+81x-81$, but the result is $-8x^3+10x^2+81x-117$, so clearly continuing this way is almost impossible by hand. Now I don't know what to do. The answer is given as $2x^2-9x+9$, so I'm not understanding something here. Thanks for your help.

Then
$$f:=2x^4+18-5x^3-5x^2=(2x-3)(x-3)(x^2+2x+2)$$
and
$$g:=4x^4-18x^3+81x-81=(2x-3)(x-3)(2x^2-9)$$
Hence, $${\rm gcd}(f,g)=(2x-3)(x-3)$$
as desired.
Note: For $f$ we have the candidate rational root are $$\pm\left\{\frac{1}{2},\frac{3}{2},\frac{9}{2}, 1,2,3,6,9,18\right\} $$ But the these we have roots $\frac{3}{2}$ and $3$, then linear factor $(2x-3)$ and $(x-3)$ so $f=(2x-3)(x-3)(x^2+2x+2)$. Similar for $g$.
