Question

Cum se simplifica aceasta fractie?
$\frac{(x^{2}+2x+4)(x^{2}+2x)+4}{(x^{2}+2x)(x^{2}+2x+3)+2}$

Answer 1

Notez: $t=x^2+2x$ .

Atunci fracția este $=\dfrac{(t+4)t+4}{t(t+3)+2}=\dfrac{t^2+4t+4}{t^2+3t+2}=\dfrac{(t+2)^2}{t^2+t+2t+2}=$

$=\dfrac{(t+2)^2}{t(t+1)+2(t+1)}=\dfrac{(t+2)^2}{(t+1)(t+2)}=\dfrac{t+2}{t+1}= \\ \\ \\ =\dfrac{x^2+2x+2}{x^2+2x+1}=\dfrac{(x+1)^2+1}{(x+1)^2}=1+\dfrac{1}{(x+1)^2}.$