[tex]f = X^4+3X^3+8X^2+mX+n \\ \\ \text{f se divide cu polinomul: }\\ \\ \Big(X-(-1+2i)\Big)\Big(X-(-1-2i)\Big)\\ \\\\ (X+1-2i)(X+1+2i) = 0 \\ (X+1)^2+4 = 0\\ X^2+2X+1+4 = 0 \\ X^2+2X+5 = 0 \Rightarrow \boxed{X^2 = -2X-5}\\ X^3+2X^2+5X = 0 \Rightarrow X^3 = -2X^2-5X\\ \Rightarrow X^3 =4X+10-5X \Rightarrow \boxed{X^3 = -X+10}\\ X^4=-2X^3-5X^2 = 2X-20+10X+25 \Rightarrow \boxed{X^4 = 12X+5}[/tex]
[tex]\text{Notam radacina polinomului care il divide pe f cu A.} \\ \\ f(A) = 0 \Rightarrow A^4+3A^3+8A^2+mA+n = 0 \\ \Rightarrow 12A+5+3( -A+10)+8(-2A-5)+mA+n = 0 \\ \Rightarrow A(12-3-16+m)+(5+30-40+n) = 0 \\ \Rightarrow (-7+m)A+(-5+n) = 0 \\ \\\Rightarrow -7+m = 0~;~~-5+n = 0\\\\\Rightarrow \boxed{m=7}~;\quad \boxed{n = 5}[/tex]
