I've basically applied the Euclidian Algorithm for computing the gcd of $f=x^4-x^3+2x-10$ and $g=x^2-x+1$ . So,
$$\begin{align}x^4-x^3+2x-10 &= (x^2-x+1)(x^2-1)+(x-9)\\ x^2-x+1&=(x-9)(x+8) +73. \end{align}$$
The solution says $\displaystyle\gcd(f,g)=g(x^3+8x^2-x-7)\left(\frac{1}{73}\right)-f(x+8)\left(\frac{1}{73}\right).$
I don't see when or how the division by $73$ was made, also how the polynomial $x^3+8x^2-x-7$ was obtained. I know how to do this for integers, but for some reason I cannot work it out with polynomials.
You get, by the Extended Euclidean algorithm, $$73=(x^3+8x^2-x-7)g(x)-(x+8)f(x) $$ there seems to be a sign error in the coefficient of $f(x)$. Then you divide both sides by $73$. The g.c.d. is defined up to a unit, but conventionally, we generally ask for a monic polynomial.