Bài 6:
a) Ta có: \(x^2-4xy+4y^2-2x+4y-35\)
\(=\left(x-2y\right)^2-2\left(x-2y\right)-35\)
\(=\left(x-2y\right)^2-7\cdot\left(x-2y\right)+5\left(x-2y\right)-35\)
\(=\left(x-2y\right)\left(x-2y-7\right)+5\left(x-2y-7\right)\)
\(=\left(x-2y-7\right)\left(x-2y+5\right)\)
b) Ta có: \(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)
\(=\left(x^2+x\right)^2+3\cdot\left(x^2+x\right)+2-12\)
\(=\left(x^2+x\right)^2+3\cdot\left(x^2+x\right)-10\)
\(=\left(x^2+x\right)^2+5\left(x^2+x\right)-2\left(x^2+x\right)-10\)
\(=\left(x^2+x\right)\left(x^2+x+5\right)-2\left(x^2+x+5\right)\)
\(=\left(x^2+x+5\right)\left(x^2+x-2\right)\)
\(=\left(x^2+x+5\right)\left(x^2+2x-x-2\right)\)
\(=\left(x^2+x+5\right)\left(x-1\right)\left(x+2\right)\)
c) Ta có: \(\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)+16\)
\(=\left(x^2+10x+16\right)\left(x^2+10x+24\right)+16\)
\(=\left(x^2+10x\right)^2+40\left(x^2+10x\right)+384+16\)
\(=\left(x^2+10x\right)^2+40\left(x^2+10x\right)+400\)
\(=\left(x^2+10x+20\right)^2\)
d) Ta có: \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x\right)^2+22\left(x^2+7x\right)+120-24\)
\(=\left(x^2+7x\right)^2+22\left(x^2+7x\right)+96\)
\(=\left(x^2+7x\right)^2+16\left(x^2+7x\right)+6\left(x^2+7x\right)+96\)
\(=\left(x^2+7x\right)\left(x^2+7x+16\right)+6\left(x^2+7x+16\right)\)
\(=\left(x^2+7x+16\right)\left(x^2+7x+6\right)\)
\(=\left(x^2+7x+16\right)\left(x+1\right)\left(x+6\right)\)
e) Ta có: \(x\left(x+4\right)\left(x+6\right)\left(x+10\right)+128\)
\(=\left(x^2+10x\right)\left(x^2+10x+24\right)+128\)
\(=\left(x^2+10x\right)^2+24\left(x^2+10x\right)+128\)
\(=\left(x^2+10x\right)^2+16\left(x^2+10x\right)+8\left(x^2+10x\right)+128\)
\(=\left(x^2+10x\right)\left(x^2+10x+16\right)+8\left(x^2+10x+16\right)\)
\(=\left(x^2+10x+16\right)\left(x^2+10x+8\right)\)
\(=\left(x+2\right)\left(x+8\right)\left(x^2+10x+8\right)\)