Square of Difference: Resulting in a quadratic equation

Fill in the blanks:

$(?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5$

$2,\text{ }1$

Solve the following equation:

$\frac{x^3+1}{(x-1)^2}=x+4$

$x=3,\frac{1}{2}$

Solve the following equation:

$(x-4)^2+3x^2=-16x+12$

$x=-1$

Solve the following equation:

$(x-5)^2-5=10+2x$

$x_1=6+\frac{\sqrt{104}}{2},\\x_2=6-\frac{\sqrt{104}}{2}$

Solve the following equation:

$(x-5)^2-5=-12+2x$

$x_1=8,x_2=4$

Solve the following system of equations:

$\begin{cases} \sqrt{x}-\sqrt{y}=\sqrt{\sqrt{61}-6} \\ xy=9 \end{cases}$

$x=\frac{\sqrt{61}}{2}-2.5$

$y=\frac{\sqrt{61}}{2}+2.5$

or

$x=\frac{\sqrt{61}}{2}+2.5$

$y=\frac{\sqrt{61}}{2}-2.5$

Solve the following equation:

$\frac{(2x-1)^2}{x-2}+\frac{(x-2)^2}{2x-1}=4.5x$

$$$-1\pm\sqrt{3}$