Find all real solutions to the inequality \( x^3 - 4x^2 + x + 6 < 0 \).

["# Find All Real Solutions to the Inequality ( x^3 - 4x^2 + x + 6 < 0 )", "If you're looking to solve the inequality ( x^3 - 4x^2 + x + 6 < 0 ), this article provides a detailed, step-by-step explanation to help you find all real solutions. Understanding how to analyze cubic inequalities is essential in algebra and calculus, and this polynomial offers a great example of combining factoring, sign analysis, and interval testing.", "---", "## Understanding the Inequality", "We want to find all real numbers ( x ) that satisfy:\n[\nx^3 - 4x^2 + x + 6 < 0\n]", "This is a strict inequality involving a cubic polynomial. To solve it, we first find where the expression equals zero—i.e., solve:\n[\nx^3 - 4x^2 + x + 6 = 0\n]\nThese roots are the critical points dividing the real number line into intervals where the sign of the expression does not change.", "---", "## Step 1: Find the Real Roots of the Cubic Equation", "We attempt to factor the cubic ( f(x) = x^3 - 4x^2 + x + 6 ).", "### Trying Rational Root Theorem\nPossible rational roots are factors of 6 over factors of 1:\n[\n\pm1, \pm2, \pm3, \pm6\n]", "Test ( x = 1 ):\n[\n1 - 4 + 1 + 6 = 4 <br/>\ne 0\n]", "Test ( x = -1 ):\n[\n-1 - 4 - 1 + 6 = 0 \quad \ ext{✓ Root found!}\n]", "So, ( x + 1 ) is a factor.", "Now perform polynomial division or use synthetic division to factor out ( (x + 1) ):", "### Polynomial Division:\nDivide ( x^3 - 4x^2 + x + 6 ) by ( x + 1 ):", "Using synthetic division with root ( -1 ):", "-1 | 1 -4 1 6\n | -1 5 -6\n ------------------\n 1 -5 6 0", "Result:\n[\nx^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6)\n]", "Now factor the quadratic:\n[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Thus, the complete factorization is:\n[\nx^3 - 4x^2 + x + 6 = (x + 1)(x - 2)(x - 3)\n]", "---", "## Step 2: Determine Where the Inequality Holds\nWe now solve:\n[\n(x + 1)(x - 2)(x - 3) < 0\n]", "The roots are ( x = -1, 2, 3 ). These divide the real line into four intervals:\n1. ( (-\infty, -1) )\n2. ( (-1, 2) )\n3. ( (2, 3) )\n4. ( (3, \infty) )", "We test the sign of the product in each interval.", "- Interval 1: ( x < -1 ) (e.g., ( x = -2 ))\n ((-)(-)(-) = -) → Negative", "- Interval 2: ( -1 < x < 2 ) (e.g., ( x = 0 ))\n ((+)(-)(-) = +) → Positive", "- Interval 3: ( 2 < x < 3 ) (e.g., ( x = 2.5 ))\n ((+)(+)(-) = -) → Negative", "- Interval 4: ( x > 3 ) (e.g., ( x = 4 ))\n ((+)(+)(+) = +) → Positive", "The inequality ( < 0 ) holds where the expression is negative:\n[\n(-\infty, -1) \cup (2, 3)\n]", "---", "## Step 3: Final Answer and Important Notes", "The real solutions to the inequality ( x^3 - 4x^2 + x + 6 < 0 ) are all ( x ) such that:\n[\nx \in (-\infty, -1) \cup (2, 3)\n]", "### Key Takeaways:\n- Factoring cubic polynomials is crucial for solving polynomial inequalities.\n- Use the roots to partition the real line into test intervals.\n- Sign analysis helps determine where the expression is negative.\n- Because the inequality is strict (( < )), endpoints are not included.", "---", "## Practical Tip: Graph the Function", "Plotting ( f(x) = (x+1)(x-2)(x-3) ) confirms that the graph dips below the x-axis on ( (-\infty, -1) ) and ( (2, 3) ), supporting our solution.", "---", "By following these steps, you can confidently solve any cubic inequality. Practice identifying factors, choosing test points, and analyzing sign changes to master this powerful technique.", "---", "Keywords:\nreal solutions to ( x^3 - 4x^2 + x + 6 < 0 ), cubic inequality, sign analysis, factoring, polynomial roots, interval testing, algebra homework help, math tips."]