\(\left(x^2+6x-1\right)^2+2x^2+x^4+2\left(x^2+6x-1\right)\left(x^2+1\right)\)

\(\left(x^2+6x-1\right)^2+2\left(x^2+6x-1\right)\left(x^2+1\right)+\left(x^2+1\right)^2-1=\left(x^2+6x-1+x^2+1\right)^2-1=\left(2x^2+6x\right)^2-1=\left(2x^2+6x-1\right)\left(2x^2+6x+1\right)\)

\(\left(x^2+6x-1\right)^2+2\left(x^2+6x-1\right)\left(x^2+1\right)+x^4+2x^2\)

\(=\left(x^2+6x-1\right)\left(x^2+6x-1+2x^2+2\right)+x^4+2x^2\)

\(=\left(x^2+6x-1\right)\left(3x^2+6x+1\right)+x^4+2x^2\)

\(=\left(2x^2+6x-1\right)\left(2x^2+6x+1\right)\)

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

\(\left(x^2+6x+9\right)^3-\left(y^2\right)^3=\left(x^2+6x+9-y^2\right)\left[\left(x^2+6x+9\right)^2+\left(x^2+6x+9\right)y^2+y^4\right]\)

\(=\left[\left(x+3\right)^2-y^2\right]\left\{\left[\left(x^2+6x+9\right)^2+2\left(x^2+6x+9\right)y^2+y^4\right]-\left(x^2+6x+9\right)y^2\right\}\)

\(=\left(x+3-y\right)\left(x+3+y\right)\left[\left(x^2+6x+9+y^2\right)^2-\left(x+3\right)^2y^2\right]\)

\(=\left(x+3-y\right)\left(x+3+y\right)\left[\left(x^2+6x+9+y^2\right)-\left(x+3\right)y\right]\left(x^2+6x+9+y^2\right)+\left(x+3\right)y\)

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

a)\(5x^2-4\left(x^2-2x+1\right)-5=5\left(x^2-1\right)-4\left(x-1\right)^2=5\left(x-1\right)\left(x+1\right)-4\left(x-1\right)^2=\left(x-1\right)\left(5x+5-4x+4\right)=\left(x-1\right)\left(x+9\right)\)

b) \(9x^2+6x-4y^2+4y=\left(9x^2+6x+1\right)-\left(4y^2-4y+1\right)=\left(3x+1\right)^2-\left(2y-1\right)^2=\left(3x+1-2y+1\right)\left(3x+1+2y-1\right)=\left(3x-2y+2\right)\left(3x+2y\right)\)

a: \(5x^2-4\left(x^2-2x+1\right)-5\)

\(=5x^2-4x^2+8x-4-5\)

\(=x^2+8x-9\)

\(=\left(x+9\right)\left(x-1\right)\)

b: \(9x^2+6x-4y^2+4y\)

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

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

\(\left(x^2+5x-3\right)\left(x^2+5x-5\right)-15=\left(x^2+5x-3\right)\left(x^2+5x-3-2\right)-15=\left(x^2+5x-3\right)^2-2\left(x^2+5x-3\right)+1-16=\left(x^2+5x-3-1\right)^2-4^2=\left(x^2+5x-4\right)^2-4^2=\left(x^2+5x-8\right)\left(x^2+5x\right)=x\left(x+5\right)\left(x^2+5x-8\right)\)

\(\left(x^2+5x-3\right)\left(x^2+5x-5\right)-15\)

\(=\left(x^2+5x\right)^2-8\left(x^2+5x\right)-15\)

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

(1 + x²)² - 4x(1 - x²)

= (1 + x²)(1 + x²) - 4x(1 - x²)

= (1 + x² - 4x)(1 + x² - 1 + x²)

= 2x²(x² - 4x + 1)

Ta có: \(\left(x^2+1\right)^2+4x\left(x^2-1\right)\)

\(=x^4+2x^2+1+4x^3-4x\)

\(=x^4+2x^3+2x^3+4x^2-2x^2-4x+1\)

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

\(\left(x^2-2x-6\right)\left(x^2-2x-11\right)+6\)

\(=\left(x^2-2x\right)^2-17\left(x^2-2x\right)+66+6\)

\(=\left(x^2-2x\right)^2-17\left(x^2-2x\right)+72\)

\(=\left(x^2-2x-8\right)\left(x^2-2x-9\right)\)

\(=\left(x-4\right)\left(x+2\right)\left(x^2-2x-9\right)\)

\(x^4+6x^3+11x^2+6x+1\)

\(=x^4+3x^3+x^2+3x^3+9x^2+3x+x^2+3x+1\)

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

\(x^4+6x^3+7x^2-6x+1\)

\(=x^4-2x^2+1+6x^3+9x^2-6x\)

\(=\left(x^2-1\right)^2+6x\left(x^2-1\right)+9x^2\)

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

x²-2x-15=(x²-5x)+(3x-15)=x(x-5)+3(x-5)=(x-5)(x+3)

\(x^2-3x-15=\left(x^2-2.\dfrac{3}{2}x+\dfrac{9}{4}\right)-\dfrac{69}{4}=\left(x-\dfrac{3}{2}\right)^2-\left(\dfrac{\sqrt{69}}{2}\right)^2\)

\(=\left(x-\dfrac{3}{2}-\dfrac{\sqrt{69}}{2}\right)\left(x-\dfrac{3}{2}+\dfrac{\sqrt{69}}{2}\right)\)

\(x^2-2x-15=\left(x-5\right)\left(x+3\right)\)