Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The function$$f'(x)$$ is

If $$f''(x)=-f(x)$$ and $$g(x)=f'(x)$$ and $$\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}$$ and given that $$\mathrm{F}(5)=5$$, then $$\mathrm{F}(10)$$ is equal to :

Find the range of value of $$t$$ for which

$$2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$