\(a.x^2-x+\dfrac{1}{4}=x^2-x+\left(\dfrac{1}{2}\right)^2=\left(x-\dfrac{1}{2}\right)^2.\)
\(b.2x^3-12x^2+24x-16=2\left(x^3-6x^2+12x-8\right)=2\left(x-2\right)^3.\)
\(c.\left(x+y\right)^3-\left(x-y\right)^3\)
\(=\left[\left(x+y\right)-\left(x-y\right)\right]\left[\left(x-y\right)^2+\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2\right]\)
\(=2y\left[\left(x+y\right)^2+\left(x-y\right)\left(x+y\right)+\left(x-y\right)^2\right]\)
\(=2y\left(3x^2+y^2\right).\)
\(d.2x^4+2x^2+2=2\left(x^4+x^2+1\right)=2\left(x^2+x+1\right)\left(x^2-x+1\right).\)
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