\(a^2-2.a.2b+4b^2-5b^2=\left(a-2b\right)^2-5b^2..\)

\(=\left(a-2b-b\sqrt{5}\right)\left(a-2b+b\sqrt{5}\right)\)

\(x^3+9x=0\)

<=> \(x\left(x^2+9\right)=0\)

<=> \(\orbr{\begin{cases}x=0\\x^2+9=0\end{cases}}\)

<=> \(\orbr{\begin{cases}x=0\\x\in\varnothing\end{cases}}\)

<=> \(x=0\)

\(9x^2-4-2\left(3x-2\right)^2=0\)

<=> \(\left(9x^2-4\right)-2\left(3x-2\right)^2=0\)

<=> \(\left[\left(3x\right)^2-2^2\right]-2\left(3x-2\right)^2=0\)

<=> \(\left(3x-2\right)\left(3x+2\right)-2\left(3x-2\right)^2=0\)

<=> \(\left(3x-2\right)\left[\left(3x+2\right)-2\left(3x-2\right)\right]=0\)

<=> \(\left(3x-2\right)\left(3x+2-6x+4\right)=0\)

<=> \(\left(3x-2\right)\left(-3x+6\right)=0\)

<=> \(\left(3x-2\right)3\left(-x+2\right)=0\)

<=> \(3\left(3x-2\right)\left(2-x\right)=0\)

<=> \(\orbr{\begin{cases}3x-2=0\\2-x=0\end{cases}}\)

<=> \(\orbr{\begin{cases}3x=2\\x=2\end{cases}}\)

<=> \(\orbr{\begin{cases}x=\frac{2}{3}\\x=2\end{cases}}\)

\(\left(x^3-x^2\right)-4x+8x-4=0\)

<=> \(\left(x^3-x^2\right)+\left(4x-4\right)=0\)

<=> \(x^2\left(x-1\right)+4\left(x-1\right)=0\)

<=> \(\left(x-1\right)\left(x^2+4\right)=0\)

<=> \(\orbr{\begin{cases}x-1=0\\x^2+4=0\end{cases}}\)

<=> \(x=1\)

\(\left(25x^2-10x\right):\left(-5x\right)-3\left(x-2\right)=4\)

<=> \(5x\left(5x-2\right)\left(-\frac{1}{5x}\right)-3\left(x-2\right)=4\)

<=> \(-\left(5x-2\right)-3\left(x-2\right)=4\)

<=> \(\left(5x-2\right)+3\left(x-2\right)=-4\)

<=> \(5x-2+3x-6=-4\)

<=> \(8x-8=-4\)

<=> \(8\left(x-1\right)=-4\)

<=> \(x-1=-\frac{1}{2}\)

<=> \(x=-\frac{3}{2}\)

\(\left(2x-1\right)^2-\left(4x^2-1\right)=0\)
\(\Leftrightarrow4x^2-4x+1-4x^2+1=0\)
\(\Leftrightarrow-4x+2=0\)
\(\Leftrightarrow x=\frac{1}{2}\)

\(\left(2x-1\right)^2-\left(4x^2-1\right)=0\)

<=> \(\left(2x-1\right)^2-\left[\left(2x\right)^2-1\right]=0\)

<=> \(\left(2x-1\right)^2-\left(2x-1\right)\left(2x+1\right)=0\)

<=> \(\left(2x-1\right)\left(2x-1-2x-1\right)=0\)

<=> \(-2\left(2x-1\right)=0\)

<=> \(2x-1=0\)

<=> \(x=\frac{1}{2}\)

x⁸ + x⁴ + 1

= x⁸ + 2x⁴ + 1 - x⁴

= [(x⁴)² + 2x⁴ + 1] - x⁴

= (x⁴ + 1)² - (x²)²

= ( x⁴ - x² + 1 ) ( x⁴ + x² + 1 )

x⁸+x+1

=(x⁸−x²)+(x²+x+1)

=x²(x⁶−1)+(x²+x+1)

=x²(x²+1)(x³−1)+(x²+x+1)

=x²(x³+1)(x−1)(x²+x+1)+(x²+x+1)

=(x²+x+1)[x²(x³+1)(x−1)+1]

=(x²+x+1)[x²(x⁴−x³+x−1)+1]

=(x²+x+1)(x⁶−x⁵+x³−x²+1)