(x-1)(x^2+4x+4)=4(x+2)

2 min read Jun 17, 2024

Solving the Equation (x-1)(x^2+4x+4) = 4(x+2)

This equation involves a quadratic expression and requires us to solve for the unknown variable 'x'. Let's break down the steps to find the solutions.

1. Expand and Simplify

Expand the left side: (x-1)(x^2+4x+4) = x(x^2+4x+4) - 1(x^2+4x+4) = x^3 + 4x^2 + 4x - x^2 - 4x - 4
Simplify: x^3 + 3x^2 - 4 = 4(x+2)
Expand the right side: x^3 + 3x^2 - 4 = 4x + 8
Move all terms to one side: x^3 + 3x^2 - 4x - 12 = 0

2. Factor the Equation

Factor by grouping: x^2(x+3) - 4(x+3) = 0
Factor out the common factor: (x^2 - 4)(x+3) = 0
Factor the difference of squares: (x-2)(x+2)(x+3) = 0

3. Solve for x

Set each factor to zero and solve:
- x - 2 = 0 => x = 2
- x + 2 = 0 => x = -2
- x + 3 = 0 => x = -3

Solution

Therefore, the solutions to the equation (x-1)(x^2+4x+4) = 4(x+2) are x = 2, x = -2, and x = -3.