12. factor the algebraic expressions completely.?

4x^2 + 34x + 42
= 2(2x^2 + 17x + 21)
= 2(2x^2 + 14x + 3x + 21)
= 2[(2x^2 + 14x) + (3x + 21)]
= 2[2x(x + 7) + 3(x + 7)]
= 2(x + 7)(2x + 3)

4x² + 34x + 42 = 0
x² + 17/2x = - 21/2
x² + 17/4x = - 21/2 + (17/4)²
x² + 17/4x = - 168/16 + 289/16
(x + 17/4)² = 121/16
x + 17/4 = 11/4

Factors:
= x + 17/4 - 11/4, = x + 6/4, = x + 3/2, = 2x + 3
= x + 17/4 + 11/4, = x + 28/4, = x + 7

Answer: 2(2x + 3)(x + 7)

Proof:
= 2(2x + 3)(x + 7)
= 2([2x * x] + [2x * 7] + [3 * x] + [3 * 7])
= 2(2x² + 14x + 3x + 21)
= 2(2x² + 17x + 21)
= 4x² + 34x + 42

a million. x^2 – 16x + sixty 3 = (x-7)(x-9) 2. 4x^2 + 34x + 40 two = 2(2x^2 + 17x + 21) = 2(2x + 3)(x + 7) 3. 16x^2 + 40x + 25 = (4x+5)(4x+5) 4. Simplify: (-4 + x2 + 2x3) – (-6 – x + 3x3) – (-6y3 + y2) = x^2 + 2x^3 - 4 - 3x^3 + x + 6 + 6y^3 - y^2 = -x^3 + x^2 +x + 6y^3 - y^2 + 2 = -x^2(x-a million) + (x-a million) + 6y^3 - y^2 + 3 = (a million-x^2)(x-a million) + y^2(y-a million) + 3 = (a million-x)(a million+x)(x-a million) + y^2(y-a million) + 3 = -(x-a million)^2(a million+x) + y^2(y-a million) + 3 Simplify the rational expressions 5. 6 - 2x / 4x -12 = 2(3-x)/4(x-3) = -2(x-3)/4(x-3) = -2/4 = -a million/2 6. 9x - 21 / 15x - 35 = 3(3x - 7) / 5(3x - 7) = 3/5

(2x+3)(2x+14)

4x^2+34x+42
4x^2+28x+6x+42
4x(x+7)+6(x+7)
(4x+6)(x+7)

This is (2x+3)(2x+14)

(2x+3)(2x+14)