\[ (12 - 4\lambda - 3\lambda + \lambda^2) - 2 = 0 \]

["Solving the Equation: [ (12 - 4\lambda - 3\lambda + \lambda^2) - 2 = 0 ]", "---", "### Introduction\nWorking with polynomial equations is a fundamental skill in algebra, enabling students and professionals alike to model real-world problems and solve complex mathematical scenarios. In this article, we will solve the equation:", "[\n(12 - 4\lambda - 3\lambda + \lambda^2) - 2 = 0\n]", "Simplifying and solving this quadratic equation step-by-step helps clarify algebraic manipulation and logical reasoning—key tools in mathematics and science fields.", "---", "### Step 1: Simplify the Left-Hand Side", "Start by simplifying the expression inside the parentheses:", "[\n12 - 4\lambda - 3\lambda + \lambda^2 - 2\n]", "Group like terms:", "- Constant terms: (12 - 2 = 10)\n- Coefficient of (\lambda): (-4\lambda - 3\lambda = -7\lambda)", "So, the equation becomes:", "[\n\lambda^2 - 7\lambda + 10 = 0\n]", "---", "### Step 2: Recognize the Standard Quadratic Form", "The simplified equation is now in standard quadratic form:", "[\n\boxed{\lambda^2 - 7\lambda + 10 = 0}\n]", "This is a second-degree equation of the type ( a\lambda^2 + b\lambda + c = 0 ), where:\n- (a = 1)\n- (b = -7)\n- (c = 10)", "---", "### Step 3: Factor the Quadratic Expression", "To solve ( \lambda^2 - 7\lambda + 10 = 0 ), look for two numbers whose product is ( a \cdot c = 1 \cdot 10 = 10 ) and sum is ( b = -7 ).", "The pair ( -5 ) and ( -2 ) satisfies:\n- ((-5) \ imes (-2) = 10)\n- ((-5) + (-2) = -7)", "Therefore, the equation factors as:", "[\n(\lambda - 5)(\lambda - 2) = 0\n]", "---", "### Step 4: Apply the Zero Product Property", "Set each factor equal to zero:", "[\n\lambda - 5 = 0 \quad \ ext{or} \quad \lambda - 2 = 0\n]", "Solving these gives:", "[\n\lambda = 5 \quad \ ext{or} \quad \lambda = 2\n]", "---", "### Conclusion and Key Takeaways", "The solutions to the equation [ (12 - 4\lambda - 3\lambda + \lambda^2) - 2 = 0 ] are:", "[\n\boxed{\lambda = 2} \quad \ ext{and} \quad \boxed{\lambda = 5}\n]", "This example demonstrates how to:", "- Simplify expressions by combining like terms\n- Rewrite and identify standard quadratic forms\n- Factor quadratics using product-sum techniques\n- Apply the zero product property for solution extraction", "Mastering such steps enhances algebra proficiency and prepares learners for advanced studies in mathematics, physics, engineering, and computer science.", "---", "### Key Phrases for SEO Optimization\ninclude:\n- “Solve quadratic equation step-by-step”\n- “Simplify polynomial equation”\n- “Factor ( \lambda^2 - 7\lambda + 10 )”\n- “Algebra solutions for students”\n- “How to solve ( \lambda^2 - 7\lambda + 10 = 0 )”\n- “Step-by-step quadratic solution”", "---", "Related Topics:\n- How to factor quadratic expressions\n- Solving first-degree and second-degree equations\n- Applications of quadratic equations in real life", "---", "By regularly practicing polynomial simplification and quadratic solving, anyone can strengthen their algebra foundation and solve complex equations with confidence."]