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 $$(4 + x_1^2)(4 + x_2^2)(4 + x_3^2)(4 + x_4^2) = \frac{125}{16}m$$. Then the value of $$m$$ is

Let $$P(x) = 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 4(x-x_1)(x-x_2)(x-x_3)(x-x_4)$$.

The expression we need is $$\prod (x_i^2 + 4)$$, which can be written as $$\prod (x_i + 2i)(x_i - 2i)$$.

Note that $$P(2i) = 4(2i-x_1)(2i-x_2)(2i-x_3)(2i-x_4)$$.

 Similarly, $$P(-2i) = 4(-2i-x_1)(-2i-x_2)(-2i-x_3)(-2i-x_4)$$.

5The product is $$\frac{P(2i) \cdot P(-2i)}{16}$$.

$$P(2i) = 4(16) + 8(-8i) - 17(-4) - 12(2i) + 9 = 64 - 64i + 68 - 24i + 9 = 141 - 88i$$.

$$P(-2i) = 141 + 88i$$.

$$P(2i)P(-2i) = 141^2 + 88^2 = 19881 + 7744 = 27625$$.

Product $$= \frac{27625}{16} = \frac{125}{16}m \implies 125m = 27625 \implies \mathbf{m = 221}$$