We need to solve 12 equations. I will solve equation number 11. The equation is $(x+1)(x-3) + (x+1)^2 = 2x(x-4)$.

1. Problem Description

We need to solve 12 equations. I will solve equation number

1

1. The equation is $(x+1)(x-3) + (x+1)^2 = 2x(x-4)$.

2. Solution Steps

First, we expand the terms on both sides of the equation.

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

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

$2x(x-4) = 2x^2 - 8x$

Substitute these back into the original equation:

$x^2 -2x -3 + x^2 + 2x + 1 = 2x^2 - 8x$

Combine like terms on the left side:

$2x^2 -2 = 2x^2 - 8x$

Subtract $2x^2$ from both sides:

$-2 = -8x$

Divide both sides by $-8$:

$x = \frac{-2}{-8}$

$x = \frac{1}{4}$

3. Final Answer

$x = \frac{1}{4}$