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**Hint:**Degree of the function can be defined as the highest power of the degree of the variable. In other words it is the sum of the raised power of the parameters involved in a function. For example, the degree of the function $f(x) = 2x + 9$ is 1 as the highest power of the variable here is 1 while the degree of the function $g(x) = xy + {x^2}y$ is 3 which comes by adding the power of the parameters which are in multiplication with each other. In this question, two functions have been given for which the degree of the product of the two functions needed to be determined. We need to first multiply the terms of the functions in the variable $x$ and then, evaluate the degree of the resultant function.

**Complete step by step solution:**Substitute $A = (3x + 6){\text{ and }}B = (2{x^2} + 3x + 4)$ in the function AB we get,

$

AB = (3x + 6)(2{x^2} + 3x + 4) \\

= 6{x^3} + 9{x^2} + 12x + 12{x^2} + 18x + 24 \\

= 6{x^3} + 21{x^2} + 30x + 24 \\

$

From the above equation, we can see that the highest power of the variable $x$ is 3.

Hence, the degree of the function AB is 3 where $A = (3x + 6){\text{ and }}B = (2{x^2} + 3x + 4)$.

Option B is correct.

**Note:**Alternatively, this question can also be solved by adding the degree of the equation $A = (3x + 6){\text{ and }}B = (2{x^2} + 3x + 4)$ as the resultant equation is the product of the two functions. Degree of the individual functions should only be added if the resultant function is in the product form of the two individual functions.

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