Question

$2(x^{2} +2x) - frac{3}{x^{2} +2x} =5$

Answer 1

$x^{2}+2x=m\\2m-frac{3}{m}-5=0\\frac{2m^{2}-5m-3 }{m}=0,mneq0\\2m^{2}-5m-3=0\\D=(-5)^{2}-4*2*(-3)=25+24=49=7^{2}\\m_{1}=frac{5+7}{4}=3\\m_{2}=frac{5-7}{4}=-0,5$

$1)x^{2}+2x=-0,5\\x^{2}+2x+0,5=0\\D=2^{2}-4*(0,5)=4-2=2\\x_{1}=frac{-2+sqrt{2} }{2}\\x_{2}=frac{-2-sqrt{2} }{2}$

$x^{2}+2x=3\\x^{2}+2x-3=0$

$D=2^{2}-4*(-3)=4+12=16=4^{2}\\x_{3}=frac{-2+4}{2} =1\\x_{4}=frac{-2-4}{2} =-3$