a) \(3x\left(x-1\right)=x^2-2x+1\)
\(\Leftrightarrow3x\left(x-1\right)=\left(x-1\right)^2\Leftrightarrow\left(x-1\right)\left(x-1-3x\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(2x+1\right)=0\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-\dfrac{1}{2}\end{matrix}\right.\)
b) \(\Leftrightarrow x^3-7x^2+14x-8=0\)
\(\Leftrightarrow x^3-2x^2-5x^2+10x+4x-8=0\)
\(\Leftrightarrow x^2\left(x-2\right)-5x\left(x-2\right)+4\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^2-5x+4\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-2\right)\left(x-4\right)=0\Leftrightarrow\left[{}\begin{matrix}x=1\\x=2\\x=4\end{matrix}\right.\)
c) \(3x^2=4x\Leftrightarrow x\left(3x-4\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{4}{3}\end{matrix}\right.\)
d) \(\Leftrightarrow x^2-6x-7=0\)
\(\Leftrightarrow x^2-6x+9-16=0\)
\(\Leftrightarrow\left(x-3\right)^2-16=0\Leftrightarrow\left(x-7\right)\left(x+1\right)=0\Leftrightarrow\left[{}\begin{matrix}x=7\\x=-1\end{matrix}\right.\)