Bài 2:
1: \(6x^3y-12x^2y^2+6xy^3\)
\(=6xy\cdot x^2-6xy\cdot2xy+6xy\cdot y^2\)
\(=6xy\left(x^2-2xy+y^2\right)=6xy\left(x-y\right)^2\)
2: \(\left(x^2+4\right)^2-16\)
\(=\left(x^2+4+4\right)\left(x^2+4-4\right)=x^2\left(x^2+8\right)\)
3: \(5x^2-5xy-10x+10y\)
=5x(x-y)-10(x-y)
=(x-y)(5x-10)
=5(x-2)(x-y)
4: \(a^3-3a+3b-b^3\)
\(=\left(a^3-b^3\right)-3\left(a-b\right)\)
\(=\left(a-b\right)\left(a^2+ab+b^2\right)-3\left(a-b\right)\)
=(a-b)(\(\left(a^2+ab+b^2-3\right)\)
5: \(x^2-2x-y^2+1\)
\(=\left(x^2-2x+1\right)-y^2\)
\(=\left(x-1\right)^2-y^2=\left(x-1-y\right)\left(x-1+y\right)\)
6: \(x^2-x-2=x^2-2x+x-2\)
=x(x-2)+(x-2)
=(x-2)(x+1)
7: \(x^4-5x^2+4\)
\(=x^4-x^2-4x^2+4\)
\(=x^2\left(x^2-1\right)-4\left(x^2-1\right)=\left(x^2-1\right)\left(x^2-4\right)\)
=(x-2)(x+2)(x-1)(x+1)
8: \(x^5-x^3-2x^2-x\)
\(=x\left(x^4-x^2-2x-1\right)\)
\(=x\left\lbrack\left(x^2\right)^2-\left(x^2+2x+1\right)\right\rbrack\)
\(=x\left\lbrack\left(x^2\right)^2-\left(x+1\right)^2\right\rbrack=x\left(x^2-x-1\right)\left(x^2+x+1\right)\)
9: \(\left(a^3-27\right)-\left(3-a\right)\left(6a+9\right)\)
\(=\left(a-3\right)\left(a^2+3a+9\right)+\left(a-3\right)\left(6a+9\right)\)
\(=\left(a-3\right)\left(a^2+3a+9+6a+9\right)=\left(a-3\right)\left(a^2+3a+6a+18\right)\)
=(a-3)(a+3)(a+6)
10: \(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z+y^2z-xy^2+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x^2-y^2\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\cdot\left(xy+z^2\right)-z\left(x-y\right)\left(x+y\right)\)
\(=\left(x-y\right)\left(xy+z^2-xz-yz\right)\)
=(x-y)[x(y-z)-z(y-z)]
=(x-y)(y-z)(x-z)
Bài 3:
1: \(\left(x+5\right)^2=\left(x+3\right)\left(x-7\right)\)
=>\(x^2+10x+25=x^2-7x+3x-21\)
=>10x+25=-4x-21
=>14x=-46
=>\(x=-\frac{46}{14}=-\frac{23}{7}\)
2: \(x^2+6x=-9\)
=>\(x^2+6x+9=0\)
=>\(\left(x+3\right)^2=0\)
=>x+3=0
=>x=-3
3: \(\left(2x+1\right)^2-4\left(x+2\right)^2=9\)
=>\(4x^2+4x+1-4\left(x^2+4x+4\right)=9\)
=>\(4x^2+4x+1-4x^2-16x-16=9\)
=>-12x-15=9
=>-12x=24
=>x=-2
4: \(\left(x+2\right)\left(x^2-2x+4\right)=15+x\left(x^2+2\right)\)
=>\(15+x^3+2x=x^3+8\)
=>2x+15=8
=>2x=-7
=>\(x=-\frac72\)
