ANSWER
One real root.
EXPLANATION
The graph of the function
[tex]f(x) = {x}^{3} + 6{x}^{2} + 12x + 8[/tex]
is shown in the attachment.
According to the Fundamental Theorem of Algebra, this function must have 3 roots including real and complex roots.
The x-intercepts gives the number of real roots.
Observe that the graph has only one intercept.
This implies that, the equation:
[tex]{x}^{3} + 6{x}^{2} + 12x + 8 = 0[/tex]
has only one real solution.
