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\(\dfrac{2x+6}{x^2-4}=\dfrac{4x+12}{2\left(x-2\right)\left(x+2\right)}=\dfrac{5}{2x-4}-\dfrac{1}{2x+4}\)
\(2xy^2+x^2y^4+1=\left(xy^2\right)^2+2xy^2\cdot1+1^2=\left(xy^2+1\right)^2\)
a, \(25x^2+5xy+\frac{1}{4}y^2=\left(5x\right)^2+2.5x.\frac{1}{2}y+\left(\frac{1}{2}y\right)^2\)
\(=\left(5x+\frac{1}{2}y\right)^2\)
b, \(9x^2+12x+4=\left(3x\right)^2+2.3x.2+2^2=\left(3x+2\right)^2\)
c, \(x^2-6x+5-y^2-4y=\left(x^2-6x+9\right)-\left(y^2+4y+4\right)\)
\(=\left(x-3\right)^2-\left(y+2\right)^2=\left(x-y-5\right)\left(x+y-1\right)\)
d, \(\left(2x-y\right)^2+4\left(x+y\right)^2-4\left(2x-y\right)\left(x+y\right)\)
\(=\left(2x-y\right)^2-2\left(2x-y\right)\left(2x+2y\right)+\left(2x+2y\right)^2\)
\(=\left(2x-y+2x+2y\right)^2=\left(4x+y\right)^2\)
\(=\dfrac{2,5x+5-0,5x+1}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{2.5\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{-0.5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{2.5}{x-2}+\dfrac{-0.5}{x+2}\)
\(=\dfrac{5}{2x-4}+\dfrac{-1}{2x+4}\)