Solve the Equation: x²+(x-2)²=2(x+1)² with Multiple Squared Terms

Let's solve the equation. First, we'll simplify the algebraic expressions using the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$

We'll apply the mentioned formula and expand the parentheses in the expressions in the equation:

$x^2+(x-2)^2=2(x+1)^2 \\ x^2+x^2-2\cdot x\cdot2+2^2=2(x^2+2\cdot x\cdot1+1^2)\\ x^2+x^2-4x+4=2(x^2+2x+1)\\ x^2+x^2-4x+4=2x^2+4x+2\\$In the final stage, we used the distributive property on the right side of the equation.

We'll continue and combine like terms, by moving terms between sides. Then we can see that the squared term cancels out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2+x^2-4x+4=2x^2+4x+2 \\ -8x=-2\hspace{8pt}\text{/}:(-8)\\ \boxed{x=\frac{2}{8}=\frac{1}{4}}$

In the final stage, we reduced the fraction that was obtained as the solution for $x$.

Therefore, the correct answer is answer A.