​$$\text{Simplify} \\\frac{8x^3 + 27y^3 - 64z^3 + 72xyz}{4x^2 + 9y^2 + 16z^2 - 6xy + 12yz + 8zx}$$​​

Correct option is B

Given:

​$$\frac{8x^{3}+27y^{3}-64z^{3}+72xyz}{4x^{2}+9y^{2}+16z^{2}-6xy+12yz+8zx}$$​​

Formula Used:

Identity:
​$$a^{3}+b^{3}+c^{3}-3abc=(a+b+c)\big(a^{2}+b^{2}+c^{2}-ab-bc-ca\big)$$​​
Solution:

Let a = 2x, b = 3y, c = -4z.

Numerator:
​$$a^{3}+b^{3}+c^{3}-3abc$$​​
​$$=8x^{3}+27y^{3}-64z^{3}+72xyz.$$​​

Denominator:
​$$a^{2}+b^{2}+c^{2}-ab-bc-ca$$​​
​$$=4x^{2}+9y^{2}+16z^{2}-6xy+12yz+8zx.$$​​
Hence the expression becomes

$$\frac{(a+b+c)\big(a^{2}+b^{2}+c^{2}-ab-bc-ca\big)}{a^{2}+b^{2}+c^{2}-ab-bc-ca}$$​​

= a + b + c

=2x + 3y - 4z.