\(ab\left(x^2+y^2\right)-xy\left(a^2+b^2\right)\)

\(=abx^2+aby^2-a^2xy-b^2xy\)

\(=\left(abx^2-b^2xy\right)-\left(a^2xy-aby^2\right)\)

\(=bx\left(ax-by\right)-ay\left(ax-by\right)\)

\(=\left(ax-by\right)\left(bx-ay\right)\)

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

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

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

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

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

f) \(=\left(5x\right)^2+20x+4=\left(5x+2\right)^2\)

\(a)x^2-4y^2=(x-2y)(x+2y)\\b)x^2-9y^2=(x-3y)(x+3y)\\c)(2x-1)^2-4y^2=(2x-1-2y)(2x-1+2y)\\d) x^2-10xy+25y^2=(x-5y)^2\\e)9x^2-6x+1=(3x-1)^2\\f)25x^2+20x+4=(5x+2)^2\)

\(x^4+x^3-20x^2-47x-15\)

\(=x^3\left(x-5\right)+6x^2\left(x-5\right)+10x\left(x-5\right)+3\left(x-5\right)\)

\(=\left(x-5\right)\left(x^3+6x^2+10x+3\right)\)

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

\(=\left(x-5\right)\left(x+3\right)\left(x^2+3x+1\right)\)

\(=x^4-5x^3+6x^3-30x^2+10x^2-50x+3x-15\\ =\left(x-5\right)\left(x^3+6x^2+10x+3\right)\\ =\left(x-5\right)\left(x^3+3x^2+3x^2+9x+x+3\right)\\ =\left(x-5\right)\left(x+3\right)\left(x^2+3x+1\right)\)

\(-25x^2\sqrt{2}+10x+4\sqrt{2}=-\sqrt{2}\left(25x^2-\dfrac{10}{\sqrt{2}}-4\right)=-\sqrt{2}.\left(\left(25x\right)^2-2.5.\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}-\dfrac{5}{2}\right)=-\sqrt{2}\left[\left(5x-\dfrac{1}{\sqrt{2}}\right)^2-\dfrac{5}{2}\right]=-\sqrt{2}.\left(5x-\dfrac{1}{\sqrt{2}}-\dfrac{\sqrt{5}}{\sqrt{2}}\right).\left(5x-\dfrac{1}{\sqrt{2}}+\dfrac{\sqrt{5}}{\sqrt{2}}\right)=-\sqrt{2}.\left(5x-\dfrac{1+\sqrt{5}}{\sqrt{2}}\right)\left(5x-\dfrac{1-\sqrt{5}}{\sqrt{2}}\right)\)

x⁴-25x²+26x-4

= (x⁴-25x²)+ (26x-4)

= ((x²)²-(5x)²)+ 2(13x-2)

= (x²-5x)(x²+5x)

a: \(x^4+25x^2+20x-4\)

\(=x^4-5x^3+2x^2+5x^3-25x^2+10x-2x^2+10x-4\)

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

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

b: \(=x^4-6x^2-x^2+9\)

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

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

c: \(=abx^2+aby^2-a^2xy-b^2xy\)

\(=\left(abx^2-b^2xy\right)+\left(aby^2-a^2xy\right)\)

\(=xb\left(ax-by\right)+ay\left(by-ax\right)\)

\(=\left(ax-by\right)\cdot\left(xb-ay\right)\)