To find the values of u, dx, and v, we need to solve the given equation and use the given information about the curve passing through the point P(2, 3) and having dy = 4 at P.

**1. Equation of the curve:**
The equation of the curve is given as y^2 = ux^3 + v.

**2. Substituting the coordinates of P(2, 3):**
Substituting the coordinates of P(2, 3) into the equation, we get:
(3)^2 = u(2)^3 + v
9 = 8u + v

**3. Differentiating the equation with respect to x:**
Differentiating the equation of the curve with respect to x, we get:
2y * dy/dx = 3u * x^2

**4. Substituting the coordinates of P(2, 3) and dy = 4 at P:**
Substituting the coordinates of P(2, 3) and dy = 4 at P into the differentiated equation, we get:
2(3) * 4 = 3u * (2)^2
24 = 12u

**5. Solving the equations:**
Now we have two equations:
9 = 8u + v
24 = 12u

Solving these two equations simultaneously, we can find the values of u and v.

Subtracting the first equation from the second equation, we get:
24 - 9 = 12u - 8u + v
15 = 4u + v

Substituting the value of 12u from the second equation into the above equation, we get:
15 = 4(2) + v
15 = 8 + v
v = 15 - 8
v = 7

Substituting the value of v into the first equation, we get:
9 = 8u + 7
2 = 8u
u = 2/8
u = 1/4

**6. Finding the values of u dx and v:**
From the above calculations, we found that u = 1/4 and v = 7.

To find the value of u dx, we can substitute the values of u and dx into the differentiated equation:
2y * dy/dx = 3u * x^2
2(3) * dx = 3(1/4) * (2)^2
6 * dx = 3/2 * 4
6 * dx = 12/2
6 * dx = 6
dx = 1

Therefore, the values of u dx and v are u dx = 1 and v = 7.

Hence, the answer is a) (u=1/4, dx=1, v=7).