Division of polynomials to find unknown coefficients The polynomial $-x^4 -x^3 + ax^2 -17x +b$, where $a$ and $b$ are constants, is denoted by $p(x)$. It is given that $x^2 + 4x + 1$ is a factor of $p(x)$.
(i) Find the values of $a$ and $b$.
(ii) With theses values of $a$ and $b$, show that the equation $p(x)=0$ has exactly two roots.

by using the long division of $\frac{-x^4-x^3+ax^2-17x+b}{x^2+4x+1}$
$$\frac{-x^4-x^3+ax^2-17x+b}{x^2+4x+1}=(-x^2+3x+a-11)+\frac{(24-4a)x+b-a+11}{x^2+4x+1}$$
the reminder should be zero
$$(24-4a)x+b-a+11=0$$
$$24-4a=0$$
$$a=6$$
$$b-a+11=0$$
$$b-6+11=0$$
$$b=-5$$ 
$$-x^2+3x+a-11=-x^2+3x+6-11=-x^2+3x-5$$
or
$$x^2-3x+5$$
the roots of this equation are not real,but the $x^2+4x+1$ has two real roots
