To solve this equation, we can simplify both sides first before trying to find an appropriate solution.

Given equation:
[tex] {2}^{ {x}^{2} + 2x} \times {3}^{ {x}^{2} + 2x } = {216}^{x + 2} [/tex]

We know that:
[216 = 2^3 \times 3^3]

Therefore, the equation can be rewritten as:
[ {2}^{ {x}^{2} + 2x} \times {3}^{ {x}^{2} + 2x } = {2^{3x+6} \times 3^{3x+6}} ]

Now we have the bases equal on both sides, we can proceed with equating the exponents:
[ {x}^{2} + 2x = 3x + 6 ]

Solving for x:
[ {x}^{2} - x - 6 = 0 ]
[ (x-3)(x+2) = 0 ]

Thus, the solutions are: x = 3 and x = -2.