$\left[\frac{1}{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}}\right]^{8}+\left[\frac{1}{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}}\right]^{8}$

After rationalise the polynomial we get

=$\left[\frac{1}{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}} \times \frac{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}}{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}}\right]^{8}$

$+\left[\frac{1}{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}} \times \frac{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}}{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}}\right]^{8}$

=$\left[\frac{\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1}}{\left(5 x^{3}+1\right)-\left(5 x^{3}-1\right)}\right]^{8}+\left[\frac{\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1}}{\left(5 x^{3}+1\right)-\left(5 x^{3}-1\right)}\right]^{8}$

$=\frac{1}{2^{8}}\left[(\sqrt{5 x^{3}+1}+\sqrt{5 x^{3}-1})^{8}+(\sqrt{5 x^{3}+1}-\sqrt{5 x^{3}-1})^{8}\right]$

=$^{8} C_{0}(\sqrt{5 x^{3}+1})^{8}+^{8} C_{2}(\sqrt{5 x^{3}+1})^{6}(\sqrt{5 x^{3}-1})^{2}$$\frac{1}{2^{8}} \cdot C_{4}(\sqrt{5 x^{3}+1})^{4}(\sqrt{5 x^{3}-1})^{4}+$

$^{\prime} C_{6}\left(\sqrt{5 x^{3}+1}^{2}(\sqrt{5 x^{3}})^{6}+^{6} C_{8}\left(y^{\sqrt{5 x^{3}-1}}\right.\right.$

$\frac{1}{2^{8}}\left[\begin{array}{l}{^{8} C_{0}\left(5 x^{3}+1\right)^{4}+^{8} C_{2}\left(5 x^{3}+1\right)^{3}\left(5 x^{3}-1\right)+8_{C_{4}}} \\ {\left(5 x^{3}+1\right)^{2}\left(5 x^{3}-1\right)^{2}+} \\ {^{8} C_{6}\left(5 x^{3}+1\right)\left(5 x^{3}-1\right)^{3}+8_{C_{4}}\left(5 x^{3}-1\right)^{4}}\end{array}\right]$

So, the degree of polynomial is $12$ , Now, coefficient of $x^{12}=\left[^{8} \mathrm{C}_{0} 5^{4}+^{8} \mathrm{C}_{2} 5^{4}+^{8} \mathrm{C}_{4} 5^{4}+^{8} \mathrm{C}_{6} 5^{4}+^{8} \mathrm{C}_{8} \mathrm{5}^{4}\right]$

$=5^{4} \times \frac{2^{8}}{2}=5^{4} \times 2^{4} \times \frac{2^{2}}{2}$

$=10^{4} \times 2^{3}=8(10)^{4}$