a) \(x^2-4x+3\)
\(=\left(x\right)^2-2\left(x\right).\left(2\right)+\left(2\right)^2-4+3\)
\(=\left(x-2\right)^2-1\)
\(=\left(x-2-1\right).\left(x-2+1\right)\)
\(=\left(x-3\right).\left(x-1\right)\)
b) \(x^2+5x+4\)
\(=\left(x\right)^2+2.\left(x\right).\left(\dfrac{5}{2}\right)+\left(\dfrac{5}{2}\right)^2-\dfrac{25}{4}+4\)
\(=\left(x+\dfrac{5}{2}\right)^2-\dfrac{25}{4}+\dfrac{16}{4}\)
\(=\left(x+\dfrac{5}{2}\right)^2-\dfrac{9}{4}\)
\(=\left(x+\dfrac{5}{2}\right)^2-\left(\dfrac{3}{2}\right)^2\)
\(=\left(x+\dfrac{5}{2}-\dfrac{3}{2}\right).\left(x+\dfrac{5}{2}+\dfrac{3}{2}\right)\)
\(=\left(x+1\right).\left(x+4\right)\)
c) \(x^2-x-6\)
\(=\left(x\right)^2-2.\left(x\right).\left(\dfrac{1}{2}\right)+\left(\dfrac{1}{2}\right)^2-\dfrac{1}{4}-6\)
\(=\left(x-\dfrac{1}{2}\right)^2-\dfrac{1}{4}-\dfrac{24}{4}\)
\(=\left(x-\dfrac{1}{2}\right)^2-\dfrac{25}{4}\)
\(=\left(x-\dfrac{1}{2}\right)^2-\left(\dfrac{5}{2}\right)^2\)
\(=\left(x-\dfrac{1}{2}-\dfrac{5}{2}\right).\left(x-\dfrac{1}{2}+\dfrac{5}{2}\right)\)
\(=\left(x-3\right).\left(x+2\right)\)