Solving the Equation (x+3)^3  x(3x+1)^2 + (2x+1)(4x^22x+1)  3x^2 = 54
This article will guide you through the process of solving the given equation. We will simplify the equation stepbystep, employing algebraic techniques to arrive at the solution.
Expanding and Simplifying the Equation

Expand the terms: Begin by expanding the powers and products in the equation:
 (x+3)^3 = x^3 + 9x^2 + 27x + 27
 x(3x+1)^2 = x(9x^2 + 6x + 1) = 9x^3 + 6x^2 + x
 (2x+1)(4x^22x+1) = 8x^3  4x^2 + 2x + 4x^2  2x + 1 = 8x^3 + 1

Substitute the expanded terms: Now substitute these expanded terms back into the original equation:
x^3 + 9x^2 + 27x + 27  (9x^3 + 6x^2 + x) + 8x^3 + 1  3x^2 = 54

Simplify by combining like terms: Combine the terms with the same powers of x:
(x^3  9x^3 + 8x^3) + (9x^2  6x^2  3x^2) + (27x  x) + (27 + 1) = 54
This simplifies to: 0 + 0 + 26x + 28 = 54
Solving for x

Isolate the variable: Move the constant term to the right side of the equation:
26x = 54  28 26x = 26

Solve for x: Divide both sides by 26 to find the value of x:
x = 26 / 26 x = 1
Conclusion
Therefore, the solution to the equation (x+3)^3  x(3x+1)^2 + (2x+1)(4x^22x+1)  3x^2 = 54 is x = 1.