Question

X(x+2)-(x+2)(x²-2x+4)-2x+4x=0

Answer 1

X^2+2x-(x^3-2x^2+4x+2x^2-4x+8)-2x+4x
x^2+2x-x^3+2x^2-4x-2x^2+4x-8-2x+4x
x^2-x^3-8+4x

Answer 2

To solve the given equation, let's break down the steps:

1. Distribute the terms within the parentheses.
2. Combine like terms on both sides of the equation.
3. Factor out common terms.
4. Solve for the variable.

Let's work through it:

Given equation:

X(x+2) - (x+2)(x²-2x+4) - 2x + 4x = 0

Step 1: Distribute terms within the parentheses

X(x) + X(2) - (x)(x²) - (x)(-2x) - (x)(4) - (2)(x²) + (2)(2x) - (2)(4) - 2x + 4x = 0

Simplify:

X² + 2X - x³ + 2x² - 4x - 2x² + 4x - 8 - 2x + 4x = 0

Step 2: Combine like terms

X² + 2X - x³ + 4x - 8 = 0

Step 3: Factor out common terms

X² + x³ + 6x - 8 = 0

Step 4: Solve for the variable

This is a cubic equation and solving cubic equations can sometimes be complex. However, I'll show you how to find possible rational roots using the Rational Root Theorem, which can help us identify potential solutions.

The Rational Root Theorem states that if a rational number p/q (where p is a factor of the constant term and q is a factor of the leading coefficient) is a root of the polynomial equation, then p is a factor of the constant term and q is a factor of the leading coefficient.

In our case, the constant term is -8 and the leading coefficient is 1. The factors of -8 are ±1, ±2, ±4, and ±8. The factors of 1 are ±1. So, the possible rational roots are:

±1, ±2, ±4, ±8

Now, you would need to use numerical methods, such as synthetic division or graphing, to find the actual solutions of the cubic equation. Keep in mind that cubic equations might have complex or irrational roots as well.

If you're looking for a numerical solution, you could use a calculator or computer software to find the approximate solutions. If you're looking for exact solutions, you might need to use more advanced algebraic techniques.