Let $\alpha $ is a root of $x^2+3x+5=0$ then express Let $\alpha $ is a root of $x^2+3x+5=0$ then express $\alpha^4 + 5\alpha^3 + 7\alpha^2 + 8\alpha -9$ as a linear expression of $\alpha $.
My Attempts:
since $\alpha $ is a root of $x^2+3x+5=0$
We have,
$$\alpha^2 + 3\alpha +5=0$$.
..

Because $$x^4+5x^3+7x^2+8x-9=$$
$$=x^2(x^2+3x+5)+2x(x^2+3x+5)-4(x^2+3x+5)+10x+11.$$
Thus, since $\alpha^2+3\alpha+5=0$, we got that our expression it's $10\alpha+11$.
Also, since a degree of the remainder should be less that $2$ we must get a linear expression of $\alpha$.
