Final answer:
The cubic equation with a repeated root at -2 and a root at 1, and containing the point (-1,6), is x^3 + 2x^2 - x - 2 = 0.
Step-by-step explanation:
The given cubic equation has a repeated root (multiplicity of 2) at -2 and a root at 1. To find the equation, we can use the factored form of a cubic equation: (x - r)(x - r)(x - s) = 0, where r is the repeated root and s is the other root. Since the multiplicity of -2 is 2, we have (x + 2)(x + 2)(x - 1) = 0.
To find the equation in standard form, we can multiply out the factors: (x + 2)(x + 2)(x - 1) = (x + 2)(x^2 - 1)(x - 1) = (x^2 + 3x + 2)(x - 1) = x^3 + 3x^2 + 2x - x^2 - 3x - 2 = x^3 + 2x^2 - x - 2 = 0.
We can now substitute the coordinate of the given point (-1, 6) into the equation to find the value of x and solve for the constant term:
6 = (-1)^3 + 2(-1)^2 - (-1) - 2 = -1 + 2 + 1 - 2 = 0.
Therefore, the equation that satisfies the given conditions is x^3 + 2x^2 - x - 2 = 0.