Identify the roots of the equation: `2x^3 + 12 x^2 - 72x - 432` State the multiplicity of each root. .

`2x^3 + 12 x^2 -72x -432 = 2x^2 (x + 6) - 72 (x + 6)`

`= (2x^2 - 72)(x + 6)`

The roots of an equation, f(x) is 'a' if, f(a) = 0.

Thus, the roots of the current equation can be calculated as:

`2x^2 - 72 = 0`

`2x^2 = 72`

`x^2 = 72/2 = 36`

`x = sqrt(36) = 6 and -6`

And another root is:

x + 6 = 0

or, x = -6

Thus the roots of the equation are 6 and -6 (appears twice). 

The equation can also be written in a simplified as: `2 (x+6)^2 (x-6)`

Since, the root x = -6 appears twice in the equation, the multiplicity of -6 is 2. Similarly, the multiplicity of x = 6 is 1.

Hope this helps.