a) \(a^2-25-2ab+b^2\)

\(=\left(a-b\right)^2-25\)

\(=\left(a-b-5\right)\left(a-b+5\right)\)

b) \(5x^2-6xy+y^2\)

\(=\left(3x\right)^2-2.3x.y+y^2-\left(2x\right)^2\)

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

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

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

c) \(2x^3-8x^2+8x\)

\(=2x^3-4x^2-4x^2+8x\)

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

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

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

d) \(5x-5y-3x^2+6xy-3y^2\)

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

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

\(=\left(x-y\right)\left[5-3\left(x-y\right)\right]\)

e) \(4x^4-9x^2\)

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

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

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

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

f) \(x^8+4\)

\(=\left(x^4\right)^2+2.x^4.2+2^2-2.x^4.2\)

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

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

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

i) \(4x^2-y^2+4x+1\)

\(=\left(2x\right)^2+2.2x+1-y^2\)

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

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

j) \(3x^2-7x+10\)

\(=3\left(x^2-\dfrac{7}{3}x+\dfrac{10}{3}\right)\)

\(=3\left(x^2-2.x.\dfrac{7}{6}+\dfrac{49}{36}-\dfrac{49}{36}+\dfrac{10}{3}\right)\)

\(=3\left[\left(x-\dfrac{7}{6}\right)^2+\dfrac{71}{36}\right]\)

g) \(x^5+x+1\)

\(=x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1\)

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

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

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

h) \(x^4+2019x^2+2018x+2019\)

\(=\left(x^4-x\right)+\left(2019x^2+2019x+2019\right)\)

\(=x\left(x^3-1\right)+2019\left(x^2+x+1\right)\)

\(=x\left(x-1\right)\left(x^2+x+1\right)+2019\left(x^2+x+1\right)\)

\(=\left(x^2+x+1\right)\left[x\left(x-1\right)+2019\right]\)

\(=\left(x^2+x+1\right)\left(x^2-x+2019\right)\)