Решить неравенство x(x+1)<2(1-2x-x^2)

Question

Решить неравенство
x(x+1)<2(1-2x-x^2)

Answer 1

x(x + 1) < 2(1 - 2x - x²)

x² + x < 2 - 4x - 2x²
x² + 2x² + x + 4x - 2 < 0
3x² +5x -2 < 0

3x² + 5x - 2 = 0

D=b² - 4ac
D=25+4*3*2
D=25+24
D=49
√D=7

x1 = (-5+7)/6 = 2/6 = 1/3
x2 = (-5 -7)/6 =-12/6=-2

3(x-1/3)(x+2) < 0
(x-1/3)(x+2) < 0

   +            -                 +
.......-2////////////////1/3..........

x∈ (-2; 1/3)

Answer 2

X(x + 1) < 2(1 - 2x - x²)
x² + x < 2 - 4x - 2x²
3x² + 5x - 2 < 0
3x² + 5x - 2 = 0
D = 25 + 2·4·3 = 49 = 7²
x₁ = (-5 - 7)/6 = -12/6 = -2
x₂ = (-5  + 7)/6 =1/3
3x² + 5x + 2 = 3(x - 1/3)(x + 2)
(x - 1/3)(x + 2) < 0
    +       -2||||||||||||-||||||||||||||||||1/3       + 
------------0-------------------------0--------------> x

Ответ: x ∈ (-2; 1/3).