a.
\(3x^2\left(a+b+c\right)+36xy\left(a+b+c\right)+108y^2\left(a+b+c\right)\)
\(=3\left(a+b+c\right)\left(x^2+12xy+36y^2\right)\)
\(=3\left(a+b+c\right)\left(x+6y\right)^2\)
b.
\(x^2-x-6=x^2+2x-3x-6=x\left(x+2\right)-3\left(x+2\right)=\left(x+2\right)\left(x-3\right)\)
c.
\(4x^2+x-5=4x^2-4x+5x-5=4x\left(x-1\right)+5\left(x-1\right)=\left(4x+5\right)\left(x-1\right)\)
d.
\(x^3-19x-30=x^3+5x^2+6x-5x^2-25x-30\)
\(=x\left(x^2+5x+6\right)-5\left(x^2+5x+6\right)=\left(x-5\right)\left(x^2+5x+6\right)\)
\(=\left(x-5\right)\left(x^2+2x+3x+6\right)=\left(x-5\right)\left[x\left(x+2\right)+3\left(x+2\right)\right]\)
\(=\left(x-5\right)\left(x+2\right)\left(x+3\right)\)
e.
\(x^4+x^2+1=x^4+2x^2+1-x^2=\left(x^2+1\right)^2-x^2\)
\(=\left(x^2+x+1\right)\left(x^2-x+1\right)\)
