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x^{2}+2x-3=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{-2±\sqrt{2^{2}-4\times 1\left(-3\right)}}{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, 2 for b, and -3 for c in the quadratic formula.
x=\frac{-2±4}{2}
Do the calculations.
x=1 x=-3
Solve the equation x=\frac{-2±4}{2} when ± is plus and when ± is minus.
\left(x-1\right)\left(x+3\right)>0
Rewrite the inequality by using the obtained solutions.
x-1<0 x+3<0
For the product to be positive, x-1 and x+3 have to be both negative or both positive. Consider the case when x-1 and x+3 are both negative.
x<-3
The solution satisfying both inequalities is x<-3.
x+3>0 x-1>0
Consider the case when x-1 and x+3 are both positive.
x>1
The solution satisfying both inequalities is x>1.
x<-3\text{; }x>1
The final solution is the union of the obtained solutions.