a: \(x^4+2010x^2+2009x+2010\)
\(=x^4+x^3+x^2-x^3-x^2-x+2010x^2+2010x+2010\)
\(=x^2\left(x^2+x+1\right)-x\left(x^2+x+1\right)+2010\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2-x+2010\right)\)
m: \(2x^2-5xy+2y^2\)
\(=2x^2-4xy-xy+2y^2\)
\(=2x\left(x-2y\right)-y\left(x-2y\right)=\left(x-2y\right)\left(2x-y\right)\)
n: \(x^5+x-1\)
\(=x^5-x^4+x^3+x^4-x^3+x^2-x^2+x-1\)
\(=x^3\left(x^2-x+1\right)+x^2\left(x^2-x+1\right)-\left(x^2-x+1\right)\)
\(=\left(x^2-x+1\right)\left(x^3+x^2-1\right)\)
p: \(2x^3-3x^2+2x-3\)
\(=x^2\left(2x-3\right)+\left(2x-3\right)\)
\(=\left(2x-3\right)\left(x^2+1\right)\)
b: \(\left(a+1\right)\left(a+3\right)\left(a+5\right)\left(a+7\right)+15\)
\(=\left(a^2+8a+7\right)\left(a^2+8a+15\right)+15\)
\(=\left(a^2+8a\right)^2+22\left(a^2+8a\right)+105+15\)
\(=\left(a^2+8a\right)^2+22\left(a^2+8a\right)+120\)
\(=\left(a^2+8a+10\right)\left(a^2+8a+12\right)\)
\(=\left(a^2+8a+10\right)\left(a+2\right)\left(a+6\right)\)
c: \(x^4+4=x^4+4x^2+4-4x^2\)
\(=\left(x^2+2\right)^2-\left(2x\right)^2\)
\(=\left(x^2+2-2x\right)\left(x^2+2+2x\right)\)
d: \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x\right)^2+22\left(x^2+7x\right)+120-24\)
\(=\left(x^2+7x\right)^2+22\left(x^2+7x\right)+96\)
\(=\left(x^2+7x+16\right)\left(x^2+7x+6\right)=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
f: \(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-zy+z^2-3xy\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xz-xy-yz\right)\)
g: \(x^3-x^2-14x+24\)
\(=x^3+4x^2-5x^2-20x+6x+24\)
\(=x^2\left(x+4\right)-5x\left(x+4\right)+6\left(x+4\right)\)
\(=\left(x+4\right)\left(x^2-5x+6\right)=\left(x+4\right)\left(x-2\right)\left(x-3\right)\)
h: \(-2x^3+8x-30=-2\left(x^3-4x+15\right)\)
\(=-2\left(x^3+3x^2-3x^2-9x+5x+15\right)\)
\(=-2\left[x^2\left(x+3\right)-3x\left(x+3\right)+5\left(x+3\right)\right]=-2\left(x+3\right)\left(x^2-3x+5\right)\)
i: \(17x^2-13x-94\)
\(=17x^2-47x+34x-94\)
=x(17x-47)+2(17x-47)
=(17x-47)(x+2)
j: \(3x^3-7x^2+17x-5\)
\(=3x^3-x^2-6x^2+2x+15x-5\)
\(=x^2\left(3x-1\right)-2x\left(3x-1\right)+5\left(3x-1\right)=\left(3x-1\right)\left(x^2-2x+5\right)\)
n: \(4x^2-4x-3=4x^2-6x+2x-3\)
=2x(2x-3)+(2x-3)
=(2x-3)(2x+1)
