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54x^{4}+27ax^{3}-16x-8a
Consider 54x^{4}+27x^{3}a-16x-8a as a polynomial over variable x.
\left(6x-4\right)\left(9x^{3}+\frac{9ax^{2}}{2}+6x^{2}+3ax+4x+2a\right)
Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 54x^{4} and n divides the constant factor -8a. One such factor is 6x-4. Factor the polynomial by dividing it by this factor.
2\left(3x-2\right)
Consider 6x-4. Factor out 2.
\frac{9x^{2}}{2}\left(2x+a\right)+3x\left(2x+a\right)+2\left(2x+a\right)
Consider 9x^{3}+\frac{9}{2}ax^{2}+6x^{2}+3ax+4x+2a. Do the grouping 9x^{3}+\frac{9ax^{2}}{2}+6x^{2}+3ax+4x+2a=\left(9x^{3}+\frac{9ax^{2}}{2}\right)+\left(6x^{2}+3ax\right)+\left(4x+2a\right), and factor out \frac{9x^{2}}{2},3x,2 in each of the groups respectively.
\left(2x+a\right)\left(\frac{9x^{2}}{2}+3x+2\right)
Factor out common term 2x+a by using distributive property.
\left(3x-2\right)\left(9x^{2}+6x+4\right)\left(2x+a\right)
Rewrite the complete factored expression. Simplify. Polynomial 9x^{2}+6x+4 is not factored since it does not have any rational roots.