To simplify the expression (\sqrt[3]{x^3 + 6x^2 + 12x + 8}), we first analyze the polynomial inside the cube root.

Step 1: Recognize the binomial cube

Recall the expansion of ((a + b)^3):
[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 ]

Compare this to the polynomial (x^3 + 6x^2 + 12x + 8):

• The first term is (x^3), so (a = x).
• The last term is (8 = 2^3), so (b = 2).

Verify the expansion:
[ (x + 2)^3 = x^3 + 3(x^2)(2) + 3(x)(2^2) + 2^3 = x^3 + 6x^2 + 12x + 8 ]

Step 2: Simplify the cube root

Since (\sqrt[3]{y^3} = y) for any real (y):
[ \sqrt[3]{(x + 2)^3} = x + 2 ]

Answer: (\boxed{x + 2})

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