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Factoring is a fundamental concept in algebra that involves breaking down a complex expression into simpler components, or "factors," that, when multiplied, give back the original expression. Understanding factoring is crucial in solving equations, especially quadratic equations. In this context, the expression \((x-5)(x+2)\) is already factored because it is expressed as a product of two binomials.

The goal of factoring is often to simplify expressions, making them easier to work with in solving equations. Later, these factors can be used in conjunction with the zero-product property, a key principle in solving polynomial equations. This property states that if a product of factors equals zero, then at least one of the factors must be zero.

To factor a quadratic polynomial, one usually looks for two numbers that multiply to give the constant term and add to give the coefficient of the middle term (in the case of quadratic trinomials). However, in the given exercise, the polynomial is already provided in its factored form, one step ahead in solving it.

When solving quadratic equations like \((x-5)(x+2) = 0\), the most efficient method is to use the zero-product property. This principle simplifies the solution process by allowing you to set each factor equal to zero separately, since if the product of any two numbers is zero, one of them must be zero.

Here's how it works in practice:

• Start by setting each factor equal to zero, giving two separate equations: \(x-5 = 0\) and \(x+2 = 0\).
• Solve these simple linear equations individually.

In this example:

• From \(x-5 = 0\), adding 5 to both sides gives \(x = 5\).
• From \(x+2 = 0\), subtracting 2 from both sides gives \(x = -2\).

It's important to solve each resulting equation carefully to ensure accuracy in finding all possible solutions to the original problem.

Error detection is an essential skill in algebraic problems solving. Even slight mistakes can lead to incorrect solutions, as seen in the original solution attempt. Initially, the solutions listed were \(x = -5\) or \(x = -2\), which were incorrect due to a simple algebraic oversight.

To effectively detect errors:

• Carefully check each step of your solution process.
• Verify calculations and ensure all operations follow algebraic rules.
• Review each factor and equation setup to catch any discrepancies early on.

In the given exercise, applying the zero-product property correctly ensures that each factor accounts for a potential solution by setting them independently to zero. By practicing meticulous error checking, students can avoid common pitfalls and deepen their understanding of solving algebraic equations.