Question

Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 \] and \[ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m.\] Then the value of \( m \) is ______.

Answer 1

Correct Answer: 221

The given polynomial can be expressed as:

\[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 4(x - x_1)(x - x_2)(x - x_3)(x - x_4). \]

Let \(x_1 = 2i\) and \(x_2 = -2i\). Substituting these values:

\[ 64 - 64i + 68 - 24i + 9 = 4(2i - x_1)(2i - x_2)(2i - x_3)(2i - x_4). \]

Simplify:

\[ 141 - 88i \quad \dots \quad (1) \]

Similarly, for \(-2i\):

\[ 64 + 64i + 68 + 24i + 9 = 4(-2i - x_1)(-2i - x_2)(-2i - x_3)(-2i - x_4). \]

Simplify:

\[ 141 + 88i \quad \dots \quad (2) \]

Using the given condition:

\[ \frac{125}{16}m = \frac{141^2 + 88^2}{16}. \]

Calculate:

\[ m = 221. \]