\(\dfrac{16+x^2}{x^2-16}=\dfrac{4^2+x^2}{x^2-4^2}=\dfrac{4^2+x^2}{\left(x-4\right)\left(x+4\right)}\)

\(\dfrac{16+x^2}{x^2-16}=\dfrac{4^2+x^2}{x^2-4^2}=\dfrac{ }{ }\)

\(\dfrac{4x+16}{x^2-16}=\dfrac{4\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}=\dfrac{4}{x-4}\)

\(-x^2-2x+8\\ =-\left(x^2-4x-2x+8\right)\\ =-\left[x\left(x-2\right)-4\left(x-2\right)\right]\\ =-\left(x-2\right)\left(x-4\right)\)

\(8x^3+12x^2y+6xy^2+y^3\\ =\left(2x\right)^3+12x^2y+6xy^2+y^3\\ =\left(2x+y\right)^3\)

\(2x^3-12x^2+18x=2x\left(x^2-6x+9\right)=2x\left(x-3\right)^2\)

\(8x^3+12x^2+6x+1\\ =\left(2x\right)^3+12x^2+6x+1^3\\ =\left(2x-1\right)^3\)

\(e)x^4-2x^4+x^2 =x^2.x^2-2x.x^2+x^2+1 =(x^2)(x^2-2x+1) =x^2(x-1)^2 \)

\(f)27y^3-x^3 =(3y)^3-x^3 =(3y-3)(9y^2+3xy+x^2)\)

a)\(\dfrac{x^3+2x}{x^3+1}+\dfrac{2x}{x^2-x+1}+\dfrac{1}{x+1}=\dfrac{x^3+2x}{\left(x+1\right)\left(x^2-x+1\right)}+\dfrac{2x}{x^2-x+1}+\dfrac{1}{x+1}\)

\(=\dfrac{x^3+2x}{\left(x+1\right)\left(x^2-x+1\right)}+\dfrac{2x\cdot\left(x+1\right)}{\left(x^2-x+1\right)\left(x+1\right)}+\dfrac{1.\left(x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\)

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

b)\(\dfrac{3\left(x+1\right)^2}{x^3-1}-\dfrac{1-x}{x^2+x+1}+\dfrac{3}{1-x}\\ =\dfrac{3\left(x+1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{\left(x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{3\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\\ =\dfrac{3x^2+6x+3+x^2-2x+1-3x^2+3x+3}{\left(x-1\right)\left(x^2+x+1\right)}\\ =\dfrac{x^2+x+12}{x^3-1}\)

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