Answer :
1) x² +(3√2)x +4
= x² + (√2 + 2√2)x + (√2)(2√2)
= (x² + √2 x) + [2√2 x + (√2)(2√2)] = x(x + √2) + (2√2)(x + √2)
= (x + √2)(x + 2√2)
2) x² + x^4 + 25 = x² + [(x²)² + 5²] = x² + (x² + 5)² - 10x² = (x² + 5)² - 9x²
= (x² + 5)² - (3x)² = (x² -3x + 5)(x² +3x + 5)
3)y³ -2y² -29y -42 = y³ +2y² -4y² -29y -42 = y³ +2y² -(4y² +29y + 42)
= (y³ +2y²) -(4y² +8y + 21y + 42) = y²(y+2) - [4y(y+2) + 21(y+2)]
= y²(y+2) - (y+2)(4y + 21) = (y+2)(y² - 4y -21) = (y+2)[(y²-7y) +(3y-21)
= (y+2)[(y(y-7) + 3(y-7)] = (y+2)(y-7)(y+3)
= x² + (√2 + 2√2)x + (√2)(2√2)
= (x² + √2 x) + [2√2 x + (√2)(2√2)] = x(x + √2) + (2√2)(x + √2)
= (x + √2)(x + 2√2)
2) x² + x^4 + 25 = x² + [(x²)² + 5²] = x² + (x² + 5)² - 10x² = (x² + 5)² - 9x²
= (x² + 5)² - (3x)² = (x² -3x + 5)(x² +3x + 5)
3)y³ -2y² -29y -42 = y³ +2y² -4y² -29y -42 = y³ +2y² -(4y² +29y + 42)
= (y³ +2y²) -(4y² +8y + 21y + 42) = y²(y+2) - [4y(y+2) + 21(y+2)]
= y²(y+2) - (y+2)(4y + 21) = (y+2)(y² - 4y -21) = (y+2)[(y²-7y) +(3y-21)
= (y+2)[(y(y-7) + 3(y-7)] = (y+2)(y-7)(y+3)