Can you help me to find the roots of 4x^3-21x^2+12x+32? with each step.?

Hello,

4𝑥³ – 21𝑥² + 12𝑥 + 32

   = 4𝑥³ – 16𝑥² – 5𝑥² + 20𝑥 – 8𝑥 + 32 ←←← Split -21𝑥² and 12𝑥

   = 4𝑥²(𝑥 – 4) – 5𝑥(𝑥 – 4) – 8(𝑥 – 4)

   = (𝑥 – 4)(4𝑥² – 5𝑥 – 8)

   = (𝑥 – 4)[(2𝑥)² – 2×(2𝑥)×(5/4) + (5/4)² – 25/16 – 128/16] ←←← Complete the square

   = (𝑥 – 4)[(2𝑥 – 5/4)² – (3√17)²/4²] ←←← Because a²–2ab+b²=(a–b)²

   = (𝑥 – 4)[2𝑥 – 5/4 – 3(√17)/4][2𝑥 – 5/4 + 3(√17)/4)]

          ←←← Because a²–b²=(a–b)(a+b)

   = (𝑥 – 4)[2𝑥 – (5 + 3√17)/4][2𝑥 – (5 – 3√17)/4]

Thus the roots, using null factor law:

   𝑥₁ = 4

   𝑥₂ = (5 + 3√17)/8

   𝑥₃ = (5 – 3√17)/8

Regards,

Dragon.Jade :-)