a) (5x-y) (25x² + 5xy + y²) = 5x * 25x² + 5x * 5xy + 5x * y² - y * 25x² - y * 5xy - y * y²
= 125x³ + 25x²y + 5xy² - 25x²y - 5xy² - y³
= 125x³ - 20x²y - y³
b) (x-3)(x² + 3x +9)-(54+x) = x * x² + x * 3x + x * 9 - 3 * x² - 3 * 3x - 3 * 9 - 54 - x
= x³ + 3x² + 9x - 3x² - 9x - 27 - 54 - x
= x³ - x - 81
c) (2x+y)(4x² -2xy + y²)- (2x - y)(4x²+2xy + y²)
= 2x * 4x² + 2x * (-2xy) + 2x * y² + y * 4x² + y * (-2xy) + y * y² - 2x * 4x² - 2x * 2xy - 2x * y² + y * 4x² + y * 2xy + y * y²
= 8x³ - 4x²y + 2xy² + 4x²y - 2xy² + y³ - 8x³ - 4x²y + 2xy² + 4x²y + 2xy² - y³
= -2xy²
d) (x+y)² + (x - y)² + (x+y)(x-y) - 3x² = (x² + 2xy + y²) + (x² - 2xy + y²) + (x² - y²) - 3x²
= x² + 2xy + y² + x² - 2xy + y² + x² - y² - 3x²
= x² + x² + x² + 2xy - 2xy + y² + y² - y² - 3x²
= 3x²
e) (x-3)³-(x-3)(x² + 3x +9) + 6(x + 1)²
= (x³ - 3x² + 9x - 27) - (x³ - 3x² + 9x - 27) + 6(x² + 2x + 1)
= x³ - 3x² + 9x - 27 - x³ + 3x² - 9x + 27 + 6x² + 12x + 6
= 6x² + 12x + 6
f) (x + y) (x² - xy + y²) + (x - y) (x² + xy + y²) - 2x³
= x * x² + x * (-xy) + x * y² + y * x² + y * (-xy) + y * y² + x * x² + x * xy + x * y² - y * x² - y * xy - y * y² - 2x³
= x³ - x²y + xy² + x³ + xy² + y³ + x³ + x²y + x²y - y³ - xy² - y³ - 2x³
= 3x³ - 2y