\(\frac{3}{x^3-1}=\frac{3}{\left(x-1\right)\left(x^2+x+1\right)}\)( ĐKXĐ : x ≠ 1 )

\(\frac{2x}{x^2+x+1}\)( ĐKXĐ : x ∈ R )

\(\frac{x}{x-1}\)( ĐKXĐ : x ≠ 1 )

MTC : ( x - 1 )( x² + x + 1 )

=> \(\frac{3}{x^3-1}=\frac{3}{\left(x-1\right)\left(x^2+x+1\right)}\)

=> \(\frac{2x}{x^2+x+1}=\frac{2x\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{2x^2-2x}{\left(x-1\right)\left(x^2+x+1\right)}\)

=> \(\frac{x}{x-1}=\frac{x\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{x^3+x^2+x}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(\frac{1}{-x^2+3x-2}=\frac{-1}{x^2-3x+2}=\frac{-1}{x^2-x-2x+2}=\frac{-1}{x\left(x-1\right)-2\left(x-1\right)}=\frac{-1}{\left(x-1\right)\left(x-2\right)}\) 

( ĐKXĐ : x ≠ 1 ; x ≠ 2 )

\(\frac{1}{x^2+5x-6}=\frac{1}{x^2-x+6x-6}=\frac{1}{x\left(x-1\right)+6\left(x-1\right)}=\frac{1}{\left(x-1\right)\left(x+6\right)}\)

( ĐKXĐ : x ≠ 1 ; x ≠ -6 )

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

( ĐKXĐ : x ≠ 1 ; x ≠ 3 )

MTC : ( x - 1 )( x - 2 )( x - 3 )( x + 6 )

=> \(\frac{-1}{\left(x-1\right)\left(x-2\right)}=\frac{-1\left(x+6\right)\left(x-3\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}=\frac{-x^2-3x+18}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}\)

=>\(\frac{1}{\left(x-1\right)\left(x+6\right)}=\frac{1\left(x-2\right)\left(x-3\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}=\frac{x^2-5x+6}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}\)

=> \(\frac{-1}{\left(x-1\right)\left(x-3\right)}=\frac{-1\left(x-2\right)\left(x+6\right)}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}=\frac{-x^2-4x+12}{\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x+6\right)}\)

\(\frac{1}{x^2-2x+1}=\frac{1}{\left(x-1\right)^2}\)( ĐKXĐ : x ≠ 1 )

\(\frac{2}{x^2+2x}=\frac{2}{x\left(x+2\right)}\)( ĐKXĐ : x ≠ 0 ; x ≠ -2 )

MTC : x( x + 2 )( x - 1 )²

Thực hiện phép tính

\(5x^2\left(4x^2-2x+5\right)\)

\(=20x^4-10x^3+25x^2\)

b) \(\left(6x^2-5\right)\left(2x+3\right)\)

\(=6x^2\left(2x+3\right)-5\left(2x+3\right)\)

\(=12x^3+18x^2-10x-15\)

a) x² + 7x + 12

= x² + 3x + 4x + 12

= ( x² + 3x ) + ( 4x + 12 )

= x( x + 3 ) + 4( x + 3 )

= ( x + 3 )( x + 4 )

b) x⁴ + 8x² + 16

= ( x² )² + 2.x².4 + 4²

= ( x² + 4 )²

c) x² + 26x + 25

= x² + x + 25x + 25

= ( x² + x ) + ( 25x + 25 )

= x( x + 1 ) + 25( x + 1 )

= ( x + 1 )( x + 25 )

a) \(x^2+7x+12\)

\(x^2+4x+3x+12\)

\(x\left(x+4\right)+3\left(x+4\right)\)

\(\left(x+4\right)\left(x+3\right)\)

b) \(x^4+8x^2+16\)

\(\left(x^2+4\right)^2\)

c) \(x^2+26x+25\)

\(x^2+x+25x+25\)

\(x\left(x+1\right)+25\left(x+1\right)\)

\(\left(x+25\right)\left(x+1\right)\)

\(2x^2+4y^2-4xy+x-4y=-\frac{5}{4}\)

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

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

\(\Leftrightarrow\hept{\begin{cases}x-\frac{1}{2}=0\\x-2y+1=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{2}\\y=\frac{3}{4}\end{cases}}}\)