Round #1076

Simplify the rational expression (x^2 - 4) / (x^2 + 5x + 6) and state any restrictions on x. Give the simplified form.

```csharp using System; class Program { static void Main() { // Define the expression string expression = "(x**2 - 4) / (x**2 + 5*x + 6)"; // Factor the numerator and denominator string factoredNumerator = "(x - 2)*(x + 2)"; string factoredDenominator = "(x + 3)*(x + 2)"; // Cancel out common factors string simplifiedExpression = "(x - 2)/(x + 3)"; // Define the restrictions string restrictions = "x != -3, x != -2"; Console.WriteLine("The simplified form of the expression is: " + simplifiedExpression); Console.WriteLine("The restrictions on x are: " + restrictions); } } ```

To simplify the rational expression (x^2 - 4) / (x^2 + 5x + 6), we first factor both the numerator and the denominator. The numerator x^2 - 4 can be factored as (x + 2)(x - 2) using the difference of squares formula. The denominator x^2 + 5x + 6 can be factored as (x + 3)(x + 2) by finding two numbers that multiply to 6 and add up to 5. So, the expression becomes ((x + 2)(x - 2)) / ((x + 3)(x + 2)). We can cancel out the common factor (x + 2) from both the numerator and the denominator, resulting in (x - 2) / (x + 3). However, we must note that x cannot be equal to -2 because that would make the denominator zero before simplification, and division by zero is undefined. Similarly, x cannot be equal to -3 in the simplified form because that would also make the denominator zero. Thus, the simplified form is (x - 2) / (x + 3), with restrictions that x cannot be -2 or -3.