x3 + 4x2 + x – 6 = 0

The equation you've given is a cubic equation, and solving it can be quite complex. However, we can try to solve it using the Rational Root Theorem, which states that any rational root of the equation, when expressed in simplest form, is a factor of the constant term divided by a factor of the leading coefficient.

The equation is: x^3 + 4x^2 + x - 6 = 0

Step 1: Identify the possible rational roots. The factors of -6 (the constant term) are ±1, ±2, ±3, ±6. The factors of 1 (the leading coefficient) are ±1. So, the possible rational roots are ±1, ±2, ±3, ±6.

Step 2: Use synthetic division or the Remainder Theorem to test the possible roots. If we plug these values into the equation, we find that x = 1 is a root of the equation.

Step 3: Use the result of the synthetic division to form a quadratic equation. After finding x = 1 as a root, we can reduce the original cubic equation to a quadratic equation: x^2 + 5x + 6 = 0

Step 4: Solve the quadratic equation. This can be factored into (x + 2)(x + 3) = 0, giving us roots x = -2 and x = -3.

So, the solutions to the equation x^3 + 4x^2 + x - 6 = 0 are x = 1, x = -2, and x = -3.