Option D: [tex](2 x+4)\left(4 x^{2}-8 x+16\right)[/tex] is the factorization of [tex]8 x^{3}+64[/tex]
Explanation:
The given expression is [tex]8 x^{3}+64[/tex]
We need to determine the factorization of the expression [tex]8 x^{3}+64[/tex]
Rewriting the expression as [tex](2x)^3+(4)^3[/tex]
Thus, the expression is of the form [tex]a^{3}+b^{3}[/tex]
The formula to determine the factorization of [tex]a^{3}+b^{3}[/tex] is given by
[tex]a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)[/tex]
where [tex]a=2x[/tex] and [tex]b=4[/tex]
Substituting the values in the formula [tex]a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)[/tex] , we have,
[tex](2x)^3+(4)^3=(2x+4)((2x)^2-(2x)(4)+(4)^2)[/tex]
Simplifying the terms, we have,
[tex](2x)^3+(4)^3=(2x+4)(4x^2-8x+16)[/tex]
Hence, the correct factorization of [tex]8 x^{3}+64[/tex] is [tex](2 x+4)\left(4 x^{2}-8 x+16\right)[/tex]
Therefore, Option D is the correct answer.
