1. **State the problem:** Factorize the expression $54x^4 + 27x^3 a - 16x - 8a$. 2. **Group terms:** Group the expression into two parts to factor by grouping: $$ (54x^4 + 27x^3 a) + (-16x - 8a) $$ 3. **Factor out the greatest common factor (GCF) from each group:** - From $54x^4 + 27x^3 a$, the GCF is $27x^3$: $$ 27x^3(2x + a) $$ - From $-16x - 8a$, the GCF is $-8$: $$ -8(2x + a) $$ 4. **Rewrite the expression using the factored groups:** $$ 27x^3(2x + a) - 8(2x + a) $$ 5. **Factor out the common binomial factor $(2x + a)$:** $$ (2x + a)(27x^3 - 8) $$ 6. **Recognize the difference of cubes in $27x^3 - 8$:** - $27x^3 = (3x)^3$ - $8 = 2^3$ 7. **Apply the difference of cubes formula:** $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$ 8. **Factor $27x^3 - 8$ as:** $$ (3x - 2)(9x^2 + 6x + 4) $$ 9. **Write the fully factored expression:** $$ (2x + a)(3x - 2)(9x^2 + 6x + 4) $$ **Final answer:** $$ \boxed{(2x + a)(3x - 2)(9x^2 + 6x + 4)} $$