First, let's expand and simplify each term on the left side of the equation:
(x+2)² = x² + 4x + 4
-x(3x+1)² = -x(9x² + 6x + 1) = -9x³ - 6x² - x
(2x+1)(4x²-2x+1) = 8x³ - 4x² + 4x + 4x² - 2x + 1 = 8x³
Now, substitute these results back into the original equation:
x² + 4x + 4 - 9x³ - 6x² - x + 8x³ = 42
Combine like terms:
9x³ - 6x² + x + 4 = 42
Rearrange the terms to set the equation equal to zero:
9x³ - 6x² + x + 4 - 42 = 0
9x³ - 6x² + x - 38 = 0
Therefore, the solution to the equation (x+2)²-x(3x+1)²+(2x+1)(4x²-2x+1)=42 is 9x³ - 6x² + x - 38 = 0.
First, let's expand and simplify each term on the left side of the equation:
(x+2)² = x² + 4x + 4
-x(3x+1)² = -x(9x² + 6x + 1) = -9x³ - 6x² - x
(2x+1)(4x²-2x+1) = 8x³ - 4x² + 4x + 4x² - 2x + 1 = 8x³
Now, substitute these results back into the original equation:
x² + 4x + 4 - 9x³ - 6x² - x + 8x³ = 42
Combine like terms:
9x³ - 6x² + x + 4 = 42
Rearrange the terms to set the equation equal to zero:
9x³ - 6x² + x + 4 - 42 = 0
9x³ - 6x² + x - 38 = 0
Therefore, the solution to the equation (x+2)²-x(3x+1)²+(2x+1)(4x²-2x+1)=42 is 9x³ - 6x² + x - 38 = 0.