Explanation:

To find the degree of the polynomial, we need to determine the highest power of x in the expression.

Expansion of the first term:
The expansion of [x(x^3 - 1)^1/2]^5 can be found by using the binomial theorem. According to the binomial theorem, the expansion of (a + b)^n is given by:

(a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n

In our case, a = x, b = (x^3 - 1)^(1/2), and n = 5. Substituting these values, the expansion becomes:

[x(x^3 - 1)^1/2]^5 = 5C0 * x^5 * (x^3 - 1)^(1/2)^0 + 5C1 * x^4 * (x^3 - 1)^(1/2)^1 + 5C2 * x^3 * (x^3 - 1)^(1/2)^2 + ... + 5C5 * x^0 * (x^3 - 1)^(1/2)^5

Simplifying each term, we get:

1 * x^5 * 1 + 5 * x^4 * (x^3 - 1)^(1/2) + 10 * x^3 * (x^3 - 1) + ... + 5 * x * (x^3 - 1)^(1/2)^4 + 1 * (x^3 - 1)^(1/2)^5

Expansion of the second term:
Similarly, the expansion of [x(x^3 - 1)^1/2]^5 can be found by using the same process. Since it is the same expression, the expansion will be identical to the first term:

[x(x^3 - 1)^1/2]^5 = 1 * x^5 * 1 + 5 * x^4 * (x^3 - 1)^(1/2) + 10 * x^3 * (x^3 - 1) + ... + 5 * x * (x^3 - 1)^(1/2)^4 + 1 * (x^3 - 1)^(1/2)^5

Degree of the polynomial:
Since both terms have the same powers of x, we can add the corresponding coefficients of each power. The highest power of x will determine the degree of the polynomial.

The coefficients of the x^5 terms in both expansions are 1 * x^5 * 1 and 1 * x^5 * 1, which add up to 2x^5. Therefore, the highest power of x in the expansion is 5.

The degree of the polynomial is 5 (option D).