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x^{2}+3x+2=0
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-3±\sqrt{3^{2}-4\times 1\times 2}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 3 for b, and 2 for c in the quadratic formula.
x=\frac{-3±1}{2}
Do the calculations.
x=-1 x=-2
Solve the equation x=\frac{-3±1}{2} when ± is plus and when ± is minus.
\left(x+1\right)\left(x+2\right)<0
Rewrite the inequality by using the obtained solutions.
x+1>0 x+2<0
For the product to be negative, x+1 and x+2 have to be of the opposite signs. Consider the case when x+1 is positive and x+2 is negative.
x\in \emptyset
This is false for any x.
x+2>0 x+1<0
Consider the case when x+2 is positive and x+1 is negative.
x\in \left(-2,-1\right)
The solution satisfying both inequalities is x\in \left(-2,-1\right).
x\in \left(-2,-1\right)
The final solution is the union of the obtained solutions.