Factor: \( (x - 4)(x + 1) = 0 \), so \( x = 4 \) (since \( x > 3 \) for log to be defined).

["# Factoring the Equation: Solving ( (x - 4)(x + 1) = 0 ) with Domain Considerations", "Understanding how to solve quadratic equations by factoring is a foundational skill in algebra. One useful example is solving the equation:\n[\n(x - 4)(x + 1) = 0\n]\nThis article explores how to correctly factor and solve this equation, along with important considerations—especially when logarithmic expressions are involved.", "---", "## Understanding the Equation", "The expression ( (x - 4)(x + 1) = 0 ) is a product of two factors equal to zero. According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero:", "[\nx - 4 = 0 \quad \ ext{or} \quad x + 1 = 0\n]", "This leads directly to:", "[\nx = 4 \quad \ ext{or} \quad x = -1\n]", "---", "## Choosing the Valid Solution in Context", "Although mathematically ( x = 4 ) and ( x = -1 ) both satisfy the original equation, not all solutions may be acceptable in real-world applications, especially when logarithms are involved.", "### Why domain matters for logarithmic functions", "Many equations in science, engineering, and advanced math require inputs to logarithmic functions (like ( \log_b(x) )) to be positive and defined. For example, ( \log_b(x) ) is only defined when ( x > 0 ). If solving a problem—say, modeling growth, probability, or signal strength—contains logarithms, then only solutions satisfying ( x > 0 ) are valid.", "In our case:\n- If ( x = -1 ), then ( \log_b(-1) ) is undefined for any positive base ( b > 1 ).\n- Thus, ( x = -1 ) is an extraneous solution in contexts demanding positivity.", "Hence, under the constraint ( x > 3 ), and given the constraint from the logarithm, the only viable solution is:\n[\nx = 4\n]", "---", "## Full Solution Steps", "1. Start with the factored form:\n[\n(x - 4)(x + 1) = 0\n]", "2. Apply the Zero Product Property:\n[\nx - 4 = 0 \quad \Rightarrow \quad x = 4\n]\n[\nx + 1 = 0 \quad \Rightarrow \quad x = -1\n]", "3. Evaluate domain restrictions:\n - For logarithmic expressions, ( x > 0 ) is required.\n - ( x = -1 ) fails this requirement and is discarded.", "4. Confirm final solution:\n Only ( x = 4 ) is valid when ( x > 3 ) (ensuring positivity) and satisfies the logarithmic condition.", "---", "## Why This Matters: Factoring in Applied Math", "Factoring isn’t just an algebraic exercise—it’s a key tool in solving equations that model physical phenomena or data analyses. When logs are involved, ignoring domain constraints leads to nonsensical or invalid results. Always:", "- Factor and solve fully\n- Check domain restrictions, especially positivity for logs\n- Discard any solutions that violate real-world constraints", "---", "## Summary", "The equation ( (x - 4)(x + 1) = 0 ) solves to:\n[\nx = 4 \quad \ ext{or} \quad x = -1\n]", "But when logarithms or positivity constraints require ( x > 0 ), only ( x = 4 ) is acceptable. Remember: valid solutions depend on the mathematical context!", "---", "Keywords: solving ( (x - 4)(x + 1) = 0 ), factor quadratic equations, domain of logarithmic functions, extraneous solutions, constraint-based solution, algebraic reasoning, mathematical domain constraints.", "---", "By mastering both factoring techniques and critical domain analysis, you ensure accurate, meaningful results—essential for success in algebra and advanced math applications."]