Solving the Equation: (x+5)(43x)(3x+2)^2+(2x+1)^3=(2x1)(4x^2+2x+1)
This article will guide you through the process of solving the given equation:
(x+5)(43x)(3x+2)^2+(2x+1)^3=(2x1)(4x^2+2x+1)
1. Expanding the Equation:
The first step is to expand the equation by multiplying out the brackets and simplifying.

Expanding (x+5)(43x):
 (x+5)(43x) = 4x  3x² + 20  15x = 3x²  11x + 20

Expanding (3x+2)²:
 (3x+2)² = (3x+2)(3x+2) = 9x² + 12x + 4

Expanding (2x+1)³:
 (2x+1)³ = (2x+1)(2x+1)(2x+1) = (4x² + 4x + 1)(2x+1) = 8x³ + 12x² + 6x + 1

Expanding (2x1)(4x²+2x+1):
 (2x1)(4x²+2x+1) = 8x³  4x² + 4x²  2x + 2x  1 = 8x³  1
Substituting the expanded terms into the equation:
3x²  11x + 20  (9x² + 12x + 4) + (8x³ + 12x² + 6x + 1) = 8x³  1
2. Simplifying the Equation:
Now, we can simplify the equation by combining like terms:
3x²  11x + 20  9x²  12x  4 + 8x³ + 12x² + 6x + 1 = 8x³  1
Combining terms:
8x³ + 0x²  17x + 17 = 8x³  1
3. Solving for x:
Notice that the 8x³ terms cancel out on both sides of the equation. This leaves us with:
17x + 17 = 1
Now, isolate x:
17x = 18
x = 18/17
Therefore, the solution to the equation is x = 18/17.
4. Verifying the Solution:
To ensure our solution is correct, we can substitute x = 18/17 back into the original equation and see if both sides are equal. This verification is left as an exercise for the reader.