a: \(x^4+3x^3+x^2+3x\)

\(=x\left(x^3+3x^2+x+3\right)\)

\(=x\left(x+3\right)\left(x^2+1\right)\)

c: \(x^2-xy-x+y\)

\(=x\left(x-y\right)-\left(x-y\right)\)

\(=\left(x-y\right)\left(x-1\right)\)

\(x-y=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)\)

a) x²+x-2
= x²-x+2x-2
= x(x-1)+2(x-1)
= (x+2)(x-1)
b) 2x²+5x+3
= 2x²+2x+3x+3
= 2x(x+1)+3(x+1)
= (2x+3)(x+1)
c) 3x²+5x-2
= 3x²+6x-1x-2
= 3x(x+2)-1(x+2)
= (3x-1)(x+2)

\(\left(x+y\right)^2+3\left(x+y\right)-10=\left[\left(x+y\right)^2+2\left(x+y\right).\dfrac{3}{2}+\dfrac{9}{4}\right]-\dfrac{49}{4}\)

\(=\left(x+y+\dfrac{3}{2}\right)^2-\dfrac{49}{4}=\left(x+y+\dfrac{3}{2}-\dfrac{7}{2}\right)\left(x+y+\dfrac{3}{2}+\dfrac{7}{2}\right)=\left(x+y-2\right)\left(x+y+5\right)\)

\(\left(x+y\right)^2+3\left(x+y\right)-10\)

\(=\left(x+y\right)^2+5\left(x+y\right)-2\left(x+y\right)-10\)

\(=\left(x+y+5\right)\left(x+y-2\right)\)

a) \(7\left(3x-2\right)+y\left(3x-2\right)=\left(3x-2\right)\left(7+y\right)\)

b) \(x\left(y-x\right)-3\left(x-y\right)=x\left(y-x\right)+3\left(y-x\right)=\left(y-x\right)\left(x+3\right)\)

c) \(x^2-6xy+9y^2=\left(x-3y\right)^2\)

a. (3x - 2)(7+y)

b. (x - y)(x - 3)

c. (x-3y)²

a. 7(3x - 2) + y(3x - 2)

= (7 + y)(3x - 2)

b. x(y - x) - 3(x - y)

= x(y - x) + 3(y - x)

= (x + 3)(y - x)

c. x² - 6xy + 9y²

= x² - 3y.x.2 + (3y)²

= (x - 3y)²

a. 3x - 3 + 5(x - 1)

= 3(x - 1) + 5(x - 1)

= (3 + 5)(x - 1)

= 8(x - 1)

b. x² - 25 + y² - 2xy

= (x² - 2xy + y²) - 25

= (x - y)² - 5²

= (x - y + 5)(x - y - 5)

c. x² + 2xy - 16a² + y²

= (x² + 2xy + y²) - 16a²

= (x + y)² - (4a)²

= (x + y + 4a)(x + y - 4a)

\(\left(x+y+z\right)^2+\left(x+y-z\right)^2-4z^2=\left(x+y+z\right)^2+\left(x+y-z-2z\right)\left(x+y-z+2z\right)=\left(x+y+z\right)^2+\left(x+y-3z\right)\left(x+y+z\right)=\left(x+y+z\right)\left(x+y+z+x+y-3z\right)=\left(x+y+z\right)\left(2x+2y-2z\right)=2\left(x+y+z\right)\left(x+y-z\right)\)

Ta có:

 (x + y + z)² + (x + y – z)² – 4z²

^{\(=\left(x+y-z\right)^2+\left(x+y-z\right)\left(x+y+3z\right)\)}

​\(=\left(x+y-z\right)\left(x+y+3z+x+y-z\right)\)​

\(\left(x+y+z\right)^2+\left(x+y-z\right)^2-4z^2\)

\(=x^2+y^2+z^2+2xy+2yz+2xz+x^2+y^2+z^2+2xy-2xz-2yz-4z^2\)

\(=2x^2+2y^2-2z^2+4xy\)

\(=2\left(x^2+2xy+y^2-z^2\right)\)

\(=2\left(x+y-z\right)\left(x+y+z\right)\)

\(A=-x-z\left(x-y\right)+y=-x-xz+zy+y=-x\left(1+z\right)+y\left(1+z\right)=\left(1+z\right)\left(y-x\right)\)

A = -(x-y)-z(x-y)=(x-y)(-1-z)=(y-x)(z+1)

A = -x - z(x - y) + y

A = -x - zx + zy + y

A = -(-x - zx + zy + y)

A = x + zx - zy - y

A = x + zx - y - zy

A = x(1 + z) - y(1 + z)

A = (x - y)(1 + z)