Solve the equation $2(x+1)^2 - (x-2)^2 = x(x-3)$ for $x$.

1. Problem Description

Solve the equation $2(x+1)^2 - (x-2)^2 = x(x-3)$ for $x$.

2. Solution Steps

First, expand the squared terms using the formula $(a+b)^2 = a^2 + 2ab + b^2$:

$(x+1)^2 = x^2 + 2x + 1$

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

Now substitute these expressions back into the original equation:

$2(x^2 + 2x + 1) - (x^2 - 4x + 4) = x(x-3)$

Expand the equation:

$2x^2 + 4x + 2 - x^2 + 4x - 4 = x^2 - 3x$

Combine like terms on the left side:

$x^2 + 8x - 2 = x^2 - 3x$

Subtract $x^2$ from both sides:

$8x - 2 = -3x$

Add $3x$ to both sides:

$11x - 2 = 0$

Add $2$ to both sides:

$11x = 2$

Divide both sides by $11$:

$x = \frac{2}{11}$

3. Final Answer

$x = \frac{2}{11}$