a) \(x^2+2x-8\\ =\left(x^2+2x+1\right)-9\\ =\left(x+1\right)^2-3^2\\ =\left(x+1-3\right).\left(x+1+3\right)\\ =\left(x-2\right).\left(x+4\right)\)
b) \(12x^2-13x+3\\ =12x^2-4x-9x+3\\ =4x\left(3x-1\right)-3\left(3x-1\right)\\ =\left(3x-1\right).\left(4x-3\right)\)
a) \(x^2+2x-8\)
\(=x^2+2x+1-9\)
\(=\left(x+1\right)^2-9\)
\(=\left(x+1+3\right)\left(x+1-3\right)\)
\(=\left(x+4\right)\left(x-2\right)\)
b) \(12x^2-13x+3\)
\(=12x^2-4x-9x+3\)
\(=4x\left(3x-1\right)-3\left(3x-1\right)\)
\(=\left(3x-1\right)\left(4x-3\right)\)
c) \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)\)
Đặt x2 + 7x + 11 = a, ta được
\(=\left(a-1\right)\left(a+1\right)-24\)
\(=a^2-1-24\)
\(=a^2-25\)
\(=\left(a-5\right)\left(a+5\right)\)
\(=\left(x^2+7x+11-5\right)\left(x^2+7x+11+5\right)\)
\(=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
d) \(x^5+x^4+1\)
\(=\left(x^5+x^4+x^3\right)+\left(x^2+x+1\right)-\left(x^3+x^2+x\right)\)
\(=x^3\left(x^2+x+1\right)+\left(x^2+x+1\right)-x\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^3-x+1\right)\)
a)\(x^2+2x-8\)
\(\Leftrightarrow x^2+4x-2x-8\)
\(\Leftrightarrow\left(x^2+4x\right)-\left(2x+8\right)\)
\(\Leftrightarrow x\left(x+4\right)-2\left(x+4\right)\)
\(\Leftrightarrow\left(x+4\right)\left(x-2\right)\)
b)\(12x^2-13x+3\)
\(\Leftrightarrow12x^2-9x-4x+3\)
\(\Leftrightarrow\left(12x^2-9x\right)-\left(4x-3\right)\)
\(\Leftrightarrow3x\left(4x-3\right)-\left(4x-3\right)\)
\(\Leftrightarrow\left(4x-3\right)\left(3x-1\right)\)
c)\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(\Leftrightarrow\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(\Leftrightarrow\left(x^2+5x+2x+10\right)\left(x^2+4x+3x+12\right)-24\)
\(\Leftrightarrow\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
Đặt \(x^2+7x+10=t\) ta có:
\(t\left(t+2\right)-24\)
\(\Leftrightarrow t^2+2t-24\)
\(\Leftrightarrow t^2+6t-4t-24\)
\(\Leftrightarrow\left(t^2+6t\right)-\left(4t+24\right)\)
\(\Leftrightarrow t\left(t+6\right)-4\left(t+6\right)\)
\(\Leftrightarrow\left(t+6\right)\left(t-4\right)\)
Thay t=\(x^2+7x+10\) ta có
\(\left(x^2+7x+10+6\right)\left(x^2+7x+10-4\right)\)
\(\Leftrightarrow\left(x^2+7x+16\right)\left(x^2+7x+6\right)\)
\(\Leftrightarrow\left(x^2+7x+16\right)\left(x^2+6x+x+6\right)\)
\(\Leftrightarrow\left(x^2+7x+16\right)\left[\left(x^2+6x\right)+\left(x+6\right)\right]\)
\(\Leftrightarrow\left(x^2+7x+16\right)\left[x\left(x+6\right)+\left(x+6\right)\right]\)
\(\Leftrightarrow\left(x^2+7x+16\right)\left(x+6\right)\left(x+1\right)\)
