The given equation is 2x^3 + 3x + k = 0. We need to find the range of values for the real number k such that the equation has two distinct real roots in the interval [0,1].

Discriminant:
To determine the nature of the roots of a cubic equation, we can look at its discriminant. For a cubic equation ax^3 + bx^2 + cx + d = 0, the discriminant is given by D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2.

Case 1: Discriminant is positive:
If the discriminant is positive, then the equation has three distinct real roots. Since we are looking for an equation with two distinct real roots, this case is not applicable.

Case 2: Discriminant is zero:
If the discriminant is zero, then the equation has a multiple root. Since we are looking for two distinct real roots, this case is also not applicable.

Case 3: Discriminant is negative:
If the discriminant is negative, then the equation has one real root and two complex conjugate roots. Since we are looking for two distinct real roots, this case is not applicable either.

Therefore, we can conclude that for the equation 2x^3 + 3x + k = 0 to have two distinct real roots in the interval [0,1], the discriminant must be zero.

Calculating Discriminant:
Substituting the given equation 2x^3 + 3x + k = 0 into the general formula for the discriminant, we get:
D = 18(2)(3)(k) - 4(3)^3(k) + (3)^2(0) - 4(2)(0)^3 - 27(2)(k)^2
Simplifying further, we have:
D = 108k - 108k^2

Setting Discriminant to Zero:
Since we want to find the range of values for k, such that the equation has two distinct real roots in the interval [0,1], we set the discriminant D to zero and solve for k:
108k - 108k^2 = 0
Dividing both sides by 108, we get:
k - k^2 = 0
Factoring out k, we have:
k(1 - k) = 0

Range of k:
The equation k(1 - k) = 0 has two roots, k = 0 and k = 1. We need to find the range of values for k in the interval [0,1].

Conclusion:
Therefore, the real number k for which the equation 2x^3 + 3x + k = 0 has two distinct real roots in the interval [0,1] lies between 0 and 1, inclusive.