Solve the following equation:

$$
5 x^{4}+20 x^{3}-40 x+17=0
$$
I. Solution: Divide both sides of the equation by 5

$$
x^{4}+4 x^{3}-8 x+3.4=0
$$

Notice that the first two terms on the left side are the same as the first two terms of $\left(x^{2}+2 x\right)^{2}$. Completing the square,

$$
x^{4}+4 x^{3}+4 x^{2}-4 x^{2}-8 x+3.4=\left(x^{2}+2 x\right)^{2}-4\left(x^{2}+2 x\right)+3.4=0
$$

This is a quadratic equation in $\left(x^{2}+2 x\right)$, from which

$$
x^{2}+2 x=\frac{4 \pm \sqrt{16-4 \cdot 3.4}}{2}=2 \pm \sqrt{4-3.4}=2 \pm \sqrt{0.6}
$$

From this, we further get

$$
x=\frac{-2 \pm \sqrt{4+8 \pm 4 \sqrt{6}}}{2}=-1 \pm \sqrt{3 \pm \sqrt{0.6}}
$$

Thus,

$$
\begin{array}{ll}
x_{1}=-1+\sqrt{3+\sqrt{0.6}} \sim 0.943 & x_{2}=-1-\sqrt{3 \pm \sqrt{0.6}} \sim-2.943 \\
x_{3}=-1+\sqrt{3-\sqrt{0.6}} \sim 0.492 & x_{4}=-1-\sqrt{3-\sqrt{0.6}} \sim-2.492
\end{array}
$$

Kálmán Bánhidy (Debrecen, Ref. g. III. o. t.)

II. Solution: (1) the left side can also be completed to a perfect fourth power:

$$
x^{4}+4 x^{3}+6 x^{2}+4 x+1-6 x^{2}-4 x-1-8 x+3.4=0
$$

or

$$
(x+1)^{4}-6 x^{2}-12 x-6+8.4=0
$$

which can be written as

$$
(x+1)^{4}-6(x+1)^{2}+8.4=0
$$

From this,

$$
(x+1)^{2}=\frac{6 \pm \sqrt{36-4 \cdot 8.4}}{2}=3 \pm \sqrt{9-8.4}=3 \pm \sqrt{0.6}
$$

or

$$
x_{1,2,3,4}=-1 \pm \sqrt{3 \pm \sqrt{0.6}}
$$

Gyula Parlagh (Kecskemét Katona József g. II. o. t.)

III. Solution: (1) adding 0.6 to both sides

$$
x^{4}+4 x^{3}-8 x+4=0.6
$$

or

$$
\left(x^{2}+2 x-2\right)^{2}=0.6
$$

from which

$$
x^{2}+2 x-2= \pm \sqrt{0.6}
$$

See Solution I.

László Fried (Bp., VIII., Széchenyi g. III o. t.)

IV. Solution: In this case, the necessary and sufficient condition derived from the result of problem 656 is satisfied for the reducibility of a fourth-degree equation to a quadratic equation.

Indeed, in this case, $a=4, b=0, e=-8$, and thus

$$
a^{2}-4 a b+8 c=4^{3}-8 \cdot 8=0
$$

Therefore, with the transformation $x=z-\frac{a}{4}=z-1$, we get the equation

$$
\begin{gathered}
(z-1)^{4}+4(z-1)^{3}-8(z-1)+3.4= \\
=z^{4}-6 z^{2}+8.4=0
\end{gathered}
$$

See Solution II.