We are given that,

We need to find the value of 5a + 4b + 3c + 2d + e.

Determinant of 3 × 3 matrix is given as,

So,

Since,

⇒ x⁴ – x³ – 12x² + 12x = ax⁴ + bx³ + cx² + dx + e

Comparing the left hand side and right hand side of the equation, we get

a = 1

b = -1

c = -12

d = 12

e = 0

Putting these values in 5a + 4b + 3c + 2d + e, we get

5a + 4b + 3c + 2d + e = 5(1) + 4(-1) + 3(-12) + 2(12) + 0

⇒ 5a + 4b + 3c + 2d + e = 5 – 4 – 36 + 24

⇒ 5a + 4b + 3c + 2d + e = 25 – 36

⇒ 5a + 4b + 3c + 2d + e = -11

Thus, the values of 5a + 4b + 3c + 2d + e is -11.