To solve the equation 5x(4+6x) = 290, we can follow these steps:

1. Distribute the 5x to both terms inside the parentheses:
20x + 30x^2 = 290

2. Rearrange the equation to form a quadratic equation:
30x^2 + 20x - 290 = 0

3. Divide the entire equation by 10 to simplify it:
3x^2 + 2x - 29 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. However, this equation does not factor easily, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = 2, and c = -29. Plugging these values into the quadratic formula, we get:

x = (-2 ± √(2^2 - 4(3)(-29))) / (2(3))
x = (-2 ± √(4 + 348)) / 6
x = (-2 ± √352) / 6

Simplifying further, we have:

x = (-2 ± √(16 * 22)) / 6
x = (-2 ± 4√22) / 6

Now, we can simplify the expression by dividing both the numerator and denominator by 2:

x = -1/3 ± 2/3√22

Therefore, the solutions to the equation 5x(4+6x) = 290 are:

x = -1/3 + 2/3√22
x = -1/3 - 2/3√22