To solve the equation 3sin(3x) - √3cos(9x) = 1 + 4sin^3(3x), we can start by simplifying the equation.

First, let's simplify the right side of the equation:
1 + 4sin^3(3x) = 1 + 4(sin(3x))^3

Next, let's simplify the left side of the equation:
3sin(3x) - √3cos(9x)

Using the trigonometric identity sin(3x) = sin(x)(3 - 4sin^2(x)), we can rewrite the left side of the equation as:
3sin(x)(3 - 4sin^2(x)) - √3cos(9x)

Expanding the equation, we get:
9sin(x) - 12sin^3(x) - √3cos(9x)

Now, let's rewrite the equation:
9sin(x) - 12sin^3(x) - √3cos(9x) = 1 + 4(sin(3x))^3

Since we have both sin(x) and cos(x) terms, we can use the trigonometric identity cos(9x) = cos^2(3x) - sin^2(3x) to rewrite the equation:
9sin(x) - 12sin^3(x) - √3(cos^2(3x) - sin^2(3x)) = 1 + 4(sin(3x))^3

Expanding the equation further, we get:
9sin(x) - 12sin^3(x) - √3cos^2(3x) + √3sin^2(3x) = 1 + 4(sin(3x))^3

Now, let's simplify the equation:
9sin(x) - 12sin^3(x) - √3cos^2(3x) + √3sin^2(3x) = 1 + 4(sin(3x))^3

Since we have both sin(x) and cos(x) terms, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
9sin(x) - 12sin^3(x) - √3(1 - sin^2(3x)) + √3sin^2(3x) = 1 + 4(sin(3x))^3

Expanding the equation further, we get:
9sin(x) - 12sin^3(x) - √3 + √3sin^2(3x) + √3sin^2(3x) = 1 + 4(sin(3x))^3

Simplifying the equation, we get:
9sin(x) - 12sin^3(x) + 2√3sin^2(3x) - √3 = 1 + 4(sin(3x))^3

Now, we have a polynomial equation in terms of sin(x) and sin(3x). To solve this equation, we can use numerical methods or approximation techniques.