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) \(x^2-5x-19\)
\(=x^2-2.x.\dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2-\left(\dfrac{5}{2}\right)^2-19\)
\(=\left(x-\dfrac{5}{2}\right)^2-\dfrac{25}{4}-19\)
\(=\left(x-\dfrac{5}{2}\right)^2-\dfrac{101}{4}\)
\(=\left(x-\dfrac{5}{2}\right)^2-\left(\dfrac{\sqrt{101}}{2}\right)^2\)
\(=\left(x-\dfrac{5}{2}-\dfrac{\sqrt{101}}{2}\right)\left(x-\dfrac{5}{2}+\dfrac{\sqrt{101}}{2}\right)\)
\(=\left(x-\dfrac{5+\sqrt{101}}{2}\right)\left(x+\dfrac{-5+\sqrt{101}}{2}\right)\)
c) \(4x^2-3x-1\)
\(=4x^2-4x+x-1\)
\(=4x\left(x-1\right)+\left(x-1\right)\)
\(=\left(x-1\right)\left(4x+1\right)\)
d) \(6x^4-11x^2+3\)
\(=6\left(x^4-\dfrac{11}{6}x^2+\dfrac{3}{6}\right)\)
\(=6\left[\left(x^2\right)^2-2.x^2.\dfrac{11}{12}+\left(\dfrac{11}{12}\right)^2-\left(\dfrac{11}{12}\right)^2+\dfrac{1}{2}\right]\)
\(=6\left[\left(x^2-\dfrac{11}{12}\right)^2-\dfrac{49}{144}\right]\)
\(=6\left[\left(x^2-\dfrac{11}{12}\right)^2-\left(\dfrac{7}{12}\right)^2\right]\)
\(=6\left[\left(x^2-\dfrac{11}{12}-\dfrac{7}{12}\right)\left(x^2-\dfrac{11}{12}+\dfrac{7}{12}\right)\right]\)
\(=6\left(x^2-\dfrac{3}{2}\right)\left(x^2-\dfrac{1}{3}\right)\)
e) \(\left(x^2+x\right)^2+3\left(x^2+x\right)+2\)
\(=\left(x^2+x\right)^2+2\left(x^2+x\right).\dfrac{3}{2}+\left(\dfrac{3}{2}\right)^2-\left(\dfrac{3}{2}\right)^2+2\)
\(=\left(x^2+x+\dfrac{3}{2}\right)^2-\dfrac{1}{4}\)
\(=\left(x^2+x+\dfrac{3}{2}\right)^2-\left(\dfrac{1}{2}\right)^2\)
\(=\left(x^2+x+\dfrac{3}{2}-\dfrac{1}{2}\right)\left(x^2+x+\dfrac{3}{2}+\dfrac{1}{2}\right)\)
\(=\left(x^2+x+1\right)\left(x^2+x+2\right)\)
f) \(x\left(x+1\right)\left(x+2\right)\left(x+3\right)+1\)
\(=\left[x\left(x+3\right)\right]\left[\left(x+1\right)\left(x+2\right)\right]+1\)
\(=\left(x^2+3x\right)\left(x^2+3x+2\right)+1\)
Đặt x2 + 3x + 1 = a, ta được
\(=\left(a-1\right)\left(a+1\right)+1\)
\(=a^2-1+1\)
\(=a^2\)
\(=\left(x^2+3x+1\right)^2\)
g) \(x^2-7xy+12y^2\)
\(=x^2-4xy-3xy+12y^2\)
\(=x\left(x-4y\right)-3y\left(x-4y\right)\)
\(=\left(x-4y\right)\left(x-3y\right)\)
h) \(x^2-2xy+y^2+3x-3y-10\)
\(=\left(x-y\right)^2+3\left(x-y\right)-10\)
\(=\left(x-y\right)\left(x-y+3\right)-10\)
Đặt x - y = a, ta được
\(=a\left(a+3\right)-10\)
\(=a^2+3a-10\)
\(=a^2-2a+5a-10\)
\(=a\left(a-2\right)+5\left(a-2\right)\)
\(=\left(a-2\right)\left(a+5\right)\)
\(=\left(x-y-2\right)\left(x-y+5\right)\)
