Solving the Equation (x+3)(x+5)/(x+2) = 0
This equation presents a common scenario in algebra where we need to find the values of 'x' that make the equation true. Let's break down the steps involved in solving it:
Understanding the Equation
The equation (x+3)(x+5)/(x+2) = 0 involves a rational expression. A rational expression is a fraction where the numerator and denominator are polynomials. To solve this type of equation, we need to consider the following:
 Zero Product Property: This property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
 Undefined Values: A rational expression is undefined when the denominator is zero. We need to exclude any values of 'x' that make the denominator zero.
Solving the Equation

Find the values that make the numerator zero:
 Set the numerator equal to zero: (x+3)(x+5) = 0
 Apply the zero product property:
 x + 3 = 0 => x = 3
 x + 5 = 0 => x = 5

Find the values that make the denominator zero:
 Set the denominator equal to zero: x + 2 = 0
 Solve for 'x': x = 2

Exclude the undefined value:
 Since the denominator cannot be zero, we exclude the value x = 2 from the possible solutions.

Check for extraneous solutions:
 We need to check if the solutions we found, x = 3 and x = 5, actually satisfy the original equation. In this case, both values work because they do not make the denominator zero.
Conclusion
Therefore, the solutions to the equation (x+3)(x+5)/(x+2) = 0 are x = 3 and x = 5. Remember to always consider the zero product property and undefined values when solving rational equations.