4. [5 points] Solve the equation $4 x^{4}+x^{2}+6 x-5 x^{2}|x+3|+9=0$.

翻译结果：
4. [5 points] Solve the equation $4 x^{4}+x^{2}+6 x-5 x^{2}|x+3|+9=0$.
Answer: $-\frac{3}{4} ; 1 ; \frac{1 \pm \sqrt{13}}{2}$.[^3]

Solution. The equation can be rewritten as $4 x^{4}+(x+3)^{2}-5 x^{2}|x+3|=0$ or $|x+3|^{2}-5 x^{2} |x+3|+4 x^{4}=0$. To factor the left side, note that it represents a quadratic trinomial in terms of $y=|x+3|$ with a discriminant equal to $\left(5 x^{2}\right)^{2}-4 \cdot 4 x^{4}=$ $9 x^{4}$. Therefore, the roots $y_{1,2}$ are $\frac{5 x^{2}+3 x^{2}}{2}$, i.e., $y_{1}=4 x^{2}$ and $y_{2}=x^{2}$, and the equation becomes $\left(|x+3|-4 x^{2}\right)\left(|x+3|-x^{2}\right)=0$.

The product is zero if and only if one of the factors is zero. From this, we get

$$
\left[\begin{array}{l}
4 x^{2} = x + 3 \\
4 x^{2} = - x - 3, \\
x^{2} + x + 3 = 0, \\
x^{2} - x - 3 = 0
\end{array} \Leftrightarrow \left[\begin{array}{l}
4 x^{2} - x - 3 = 0, \\
x^{2} - x - 3 = 0
\end{array} \Leftrightarrow \left[\begin{array}{l}
x=\frac{1 \pm \sqrt{13}}{2} \\
x=-0.75 \\
x=1
\end{array}\right.\right.\right.
$$