305. Solve the equation:

$$
10 x^{4}+3 x^{3}+5 x^{2}+5 x+8=0
$$
$\triangle$ Let's make the equation reduced:

$$
x^{4}+\frac{3}{10} x^{3}+\frac{1}{2} x^{2}+\frac{1}{2} x+\frac{4}{5}=0
$$

Complete the sum $x^{4}+\frac{3}{10} x^{3}$ to the square of a sum:

$$
\left(x^{2}+\frac{3}{20} x\right)^{2}+\left(\frac{191}{400} x^{2}+\frac{1}{2} x+\frac{4}{5}\right)=0
$$

The discriminant of the quadratic trinomial in the second set of parentheses is negative, so the trinomial maintains a constant sign for all $x$, and specifically a positive one. Therefore, the entire left side of the equation is positive; hence, the equation has no solutions.

Answer: no solutions.