a) (x + y)² - 2(x + y) + 1 

= (x + y)² - 2.1.(x + y) + 1

= (x + y - 1)²

b) x³ + 1 - x² - x

= (x³ - x²) - (x - 1) 

= x²(x - 1) - (x - 1)

= (x² - 1)(x - 1) = (x - 1)(x + 1)(x - 1) = (x - 1)²(x + 1) 

c) 27x³ - 0,001 

= \(\left(3x\right)^3-\frac{1}{1000}=\left(3x\right)^3-\left(\frac{1}{10}\right)^3=\left(3x-\frac{1}{10}\right)\left(9x^2+\frac{3}{10}x+\frac{1}{100}\right)\)

d) 125x³ - 1 =(5x)³ - 1 = (5x - 1)(25x² + 5x + 1) 

e) (x² + 4)² - 16x² 

= (x² + 4)² - (4x)²

= (x² - 4x + 4)(x² + 4x + 4) 

= (x - 2)²(x + 2)²

= [(x - 2)(x + 2)]²

a.\(\left(x+y\right)^2-2\left(x+y\right)+1\)

\(=\left(x+y\right)^2-2.\left(x+y\right).1+1^2\)

\(=\left[\left(x+y\right)-1\right]^2\)

\(=\left(x+y-1\right)^2\)

b.\(x^3+1-x^2-x\)

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

\(=x^2\left(x-1\right)-\left(x-1\right)\)

\(=\left(x-1\right)\left(x^2-1\right)\)

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

c.\(27x^3-0,001\)

\(=27x^3-\frac{1}{1000}\)

\(=\left(3x\right)^3-\left(\frac{1}{10}\right)^3\)

\(=\left(3x-\frac{1}{10}\right)\left(9x^2+0,3x+\frac{1}{100}\right)\)

d,\(125x^3-1\)

\(=\left(5x\right)^3-1^3\)

\(=\left(5x-1\right)\left(25x^2+5x+1\right)\)

e.\(\left(x^2+4\right)^2-16x^2\)

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

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

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

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

\(=\dfrac{x^4+10x^3+29x^2+10x^3+100x^2+290x-4x^2-40x-116+2224}{x^2+10x+29}\)

\(=x^2+10x-4+\dfrac{2224}{x^2+10x+29}\)

\(A=5x^3-125x=5x\left(x-5\right)\left(x+5\right)\)

\(B=x^3-8+\left(x-2\right)\left(5x+4\right)\)

\(=\left(x-2\right)\left(x^2+2x+4+5x+4\right)\)

\(=\left(x-2\right)\left(x+2\right)\left(x+4\right)\)

b, ( x + 4) ^2 - 25 = ( x + 4)^2 - 5^2 = ( x + 4 - 5)( x + 4 + 5) = ( x - 1)( x+9)

\(a,=\left(x^2-y^2\right)\left(x^2+y^2\right)=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\\ b,=\left(x-\sqrt{3}y\right)\left(x+\sqrt{3}y\right)\\ c,=\left[3x-2y-2\left(x+y\right)\right]\left[3x-2y+2\left(x+y\right)\right]\\ =5x\left(x-4y\right)\\ d,=\left[3\left(x-y\right)-2\left(x+y\right)\right]\left[3\left(x-y\right)+2\left(x+y\right)\right]\\ =\left(3x-3y-2x-2y\right)\left(3x-3y+2x+2y\right)\\ =\left(x-5y\right)\left(5x-y\right)\\ f,=\left(x+3\right)\left(x^2-3x+9\right)\\ g,=\left(3x-0,1\right)\left(9x^2+0,3x+0,01\right)\\ h,=\left(5x-1\right)\left(25x^2+5x+1\right)\)

\(a)x^4-y^4=(x^2-y^2)(x^2+y^2)=(x-y)(x+y)(x^2+y^2)\\ b)x^2-3y^2=\\ c)(3x-2y)^2-4(x+y)^2=(3x-2y)^2-[2(x+y)]^2\\=(3x-2y+2x+2y)(3x-2y-2x-2y)=5x(x-4y)\\ d)9(x-y)^2-4(x+y)^2=[3(x-y)]^2-[2(x+y)]^2=(3x-3y+2x+2y)(3x-3y-2x-2y)\\=(5x-y)(x-5y)\\ f)x^3+27=(x+3)(x^2-3x+9)\\ g)27x^3-0,001=(3x-0,1)(9x+0,3x+0,01)\\ h)125x^3-1=(5x-1)(25x^2+5x+1)\)

a) \(x^2-5x=0\)

\(x\left(x-5\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x=0\\x-5=0\end{cases}\Rightarrow\orbr{\begin{cases}x=0\\x=5\end{cases}}}\)

b) \(x^4=125x\)

\(x^4-125x=0\)

\(x\left(x^3-125\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x=0\\x^3-125=0\end{cases}\Rightarrow\orbr{\begin{cases}x=0\\x=5\end{cases}}}\)

c) \(3^x+3^{x+2}=90\)

\(3^x\left(1+3^2\right)=90\)

\(3^x\cdot10=90\)

\(3^x=9=3^2\)

\(\Rightarrow x=2\)

a) x² - 5x  = 0

x.(x-5) = 0

=> x = 0

x-5 = 0 => x = 5

KL:...

b) x⁴ = 125x

=> x³ = 125 = 5³

=> x = 5

c) 3^x + 3^x+2 = 90

3^x + 3^x.9 = 90

3^x.(1+9) = 90

3^x.10 = 90

3^x = 9 = 3²

=> x = 2

     \(x^2-5x=0\)

\(\Rightarrow x\left(x-5\right)=0\Rightarrow\orbr{\begin{cases}x=0\\x=5\end{cases}}\)

     \(x^4=125x\)

\(\Rightarrow x^4-125x=0\)

\(\Rightarrow x\left(x^3-125\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x=0\\x^3=125\end{cases}\Rightarrow\orbr{\begin{cases}x=0\\x^3=5^3\end{cases}\Rightarrow}}\orbr{\begin{cases}x=0\\x=5\end{cases}}\)

     \(3^x+3^{x+2}=90\)

\(\Rightarrow3^x+3^x.3^2=90\)

\(\Rightarrow3^x\left(3^2+1\right)=90\)

\(\Rightarrow3^x.10=90\)

\(\Rightarrow3^x=9\Rightarrow3^x=3^2\Rightarrow x=2\)

(125x³ - 1) : (25x² + 5x + 1)

= (5x - 1)(25x² + 5x + 1) : (25x² + 5x + 1)

= 5x - 1