Inspired by Parth Lohomi

Divide both sides by $45$ :

$x^4+\dfrac{4}{45}x-\dfrac{309}{5}=0$

Now, try to factor that into the form $(x^2+px+q)(x^2-px+r)=0$ . Expand and compare the coefficients:

$\quad \begin{aligned} q+r=p^2 &\quad ...(1) \\ r-q=\dfrac{4}{45p} &\quad ...(2) \\ qr=-\dfrac{309}{5} &\quad ...(3) \end{aligned}$

From $(1)$ and $(2)$ solve for $p$ and $q$ :

$q=\dfrac{p^2-\dfrac{4}{45p}}{2}$ , $r=\dfrac{p^2+\dfrac{4}{45p}}{2}$

Substitute in $(3)$ , expand, clear denominators and simplify:

$p^6+\dfrac{1236}{5}p^2-\dfrac{16}{2025}=0$

Now, compare that equation with the identity $(u+v)^3-3uv(u+v)-(u^3+v^3)=0$ . Another system will be formed:

$\quad \begin{aligned} p^2=u+v &\quad ...(4) \\3uv=-\dfrac{1236}{5} &\quad ...(5) \\ u^3+v^3=\dfrac{16}{2025} &\quad ...(6) \end{aligned}$

Divide $(5)$ by $3$ and cube both sides:

$u^3v^3=-\dfrac{69934528}{125}$

Now, make an equation in $z$ with roots $u$ and $v$ :

$(z-u^3)(z-v^3)=0 \Longrightarrow z^2-(u^3+v^3)z+u^3v^3=0$

$z^2-\dfrac{16}{2025}z-\dfrac{69934528}{125}=0$

Solve it using the quadratic formula:

$z=\dfrac{8 \pm 16\sqrt{8961727309}}{2025}$

So:

$u=\sqrt[3]{\dfrac{8 + 16\sqrt{8961727309}}{2025}}$

$v=\sqrt[3]{\dfrac{8 - 16\sqrt{8961727309}}{2025}}$

$p^2=u+v=\sqrt[3]{\dfrac{8 + 16\sqrt{8961727309}}{2025}}+\sqrt[3]{\dfrac{8 - 16\sqrt{8961727309}}{2025}}$

From here, let's work only with approximations:

$p=\sqrt{\sqrt[3]{\dfrac{8 + 16\sqrt{8961727309}}{2025}}+\sqrt[3]{\dfrac{8 - 16\sqrt{8961727309}}{2025}}}$

$p \approx 0.005653576127$

Obtain $q$ and $r$ :

$q \approx -7.86128162$ , $r \approx 7.861313583$

So, our original equation factors as:

$(x^2+0.005653576127x-7.86128162)(x^2-0.005653576127+7.861313583)=0$

By solving each factor with the quadratic formula we get:

$x_1 \approx 2.800972351$

$x_2 \approx -2.806625927$

$x_{3,4} \approx 0.002826788063 \pm 2.803801789i$

The smallest one is $x_2=\boxed{-2.806625927}$ .

With a graphing calculator TI-83 PLUS and window, Xmin=-3, Xmax=- 2.75::::: Ymin=-20, Ymax=20.
2nd TRACE (CALC),2 (ZERO) enter. Adjest the cursor so Y is as near 0 as possible.
For Y=.2039, we get x= $\boxed{ -2.806677 }$