This is an example Complex equations problem, everything is well understood except --(ii) in the below solution. Please can anyone explain, how anyone could have guessed the expansion in (ii) of the polynomial equation using (i).
$$\text{if } x= -5 + 2\sqrt{-4 } \text{ ; find the value of : } x^4+ 9x^3+35x^2-x+4$$
$$\text{Solution: } x +5 = 4i$$ $$\Rightarrow (x+5)^2= 16i^2$$ $$\Rightarrow (x+5)^2= -16$$ $$\Rightarrow x^2+10x+25= -16$$ $$\Rightarrow x^2+10x+41= 0 \Rightarrow(i)$$
$$\text{now, } \ x^4+ 9x^3+35x^2-x+4 $$ $$= x^2(x^2+10x+41)-x(x^2+10x+41)+4(x^2+10x+41)-160 \Rightarrow(ii)$$ $$= x^2(0)-x(0)+4(0)-160$$ $$=-160$$
$$\text{Therefore, the value of the polynomial for } x=-5 + 2\sqrt{-4 } \text{ is } -160$$
Step ii was done by long division: the polynomial $x^4 + 9x^3 + \ldots$ was divided by $x^2 + 10 x + 41$ to get $x^2 - x + 4$, with a remainder of $-160$.