Solving the Inequality: (x+1)(x3)² > 0
This problem involves solving a polynomial inequality. Let's break it down step by step:
1. Finding the Critical Points
The critical points are the values of x where the expression equals zero or is undefined.

Setting the expression equal to zero: (x + 1)(x  3)² = 0 This gives us x = 1 and x = 3.

Checking for undefined values: The expression is defined for all real values of x.
2. Creating a Sign Chart
We'll use a sign chart to determine the intervals where the expression is positive or negative.
Interval  x + 1  (x  3)²  (x + 1)(x  3)² 

x < 1    +   
1 < x < 3  +  +  + 
x > 3  +  +  + 
Explanation:
 x + 1: This factor is negative for x < 1 and positive for x > 1.
 (x  3)²: This factor is always positive because it is squared.
 (x + 1)(x  3)²: The sign of the product is determined by the signs of the individual factors.
3. Interpreting the Results
From the sign chart, we can see that:
 (x + 1)(x  3)² > 0 when 1 < x < 3 or x > 3
Solution
Therefore, the solution to the inequality (x + 1)(x  3)² > 0 is:
x ∈ (1, 3) ∪ (3, ∞)
In interval notation, the solution is x belongs to the open interval from 1 to 3, excluding 3, and the open interval from 3 to positive infinity.