1) 5x²y² + 15x²y - 30xy²
= 5xy(xy + 3x - 6y)
2) (x - 2)(x - 3) + 4 - x²
= (x - 2)(x - 3) - (x² - 4)
= (x - 2)(x - 3) - (x - 2)(x + 2)
= (x - 2)(x - 3 - x - 2)
= -5(x - 2)
3) x² - 7x + 12
= x² - 3x - 4x + 12
= (x² - 3x) - (4x - 12)
= x(x - 3) - 4(x - 3)
= (x - 3)(x - 4)
4) x³ - 2x²y + xy² - 9x
= x(x² - 2xy + y² - 9)
= x[(x² - 2xy + y²) - 3²]
= x[(x - y)² -3²]
= x(x - y - 3)(x - y + 3)
5) x² - 25 + y² + 2xy
= (x² + 2xy + y²) - 25
= (x + y)² - 5²
= (x + y - 5)(x + y + 5)
6) x² - x - 12
= x² + 3x - 4x - 12
= (x² + 3x) - (4x + 12)
= x(x + 3) - 4(x + 3)
= (x + 3)(x - 4)
7) 5x² + 5xy - x - y
= (5x² + 5xy) - (x + y)
= 5x(x + y) - (x + y)
= (x + y)(5x - 1)
8) 2xy - x² - y² + 16
= -(x² - 2xy + y² - 16)
= -[(x² - 2xy + y²) - 16]
= -[(x - y)² - 4²]
= -(x - y - 4)(x - y + 4)
9) (y² + y)² - 9y² - 9y + 20
= (y² + y)² - 9(y² + y) + 20
= (y² + y)² - 4(y² + y) - 5(y² + y) + 20
= [(y² + y)² - 4(y² + y)] - [5(y² + y) - 20]
= (y² + y)(y² + y - 4) - 5(y² + y - 4)
= (y² + y - 4)(y² + y - 5)
10) 12y(2x - 5) + 6xy(5 - 2x)
= 12y(2x - 5) - 6xy(2x - 5)
= 6y(2x - 5)(2 - x)
11) 16x² + 24x - 8xy - 6y + y²
= (16x² - 8xy + y²) + (24x - 6y)
= (4x - y)² + 4(4x - y)
= (4x - y)(4x - y + 4)
12) x⁴ - x² + 2xy - y²
= x⁴ - (x² - 2xy + y²)
= (x²)² - (x - y)²
= (x² - x + y)(x² + x - y)
