4. (20 points) For what values of the parameter \( a \) does the equation \(\left|\frac{-4 x^{4}-(6 a+10) x^{3}+(16-4 a) x^{2}-\left(6 a^{2}-14 a-40\right) x}{\left(4-x^{2}-a\right)(3 a+2 x+5)}\right|=\sqrt{a^{2}-2 a+1}\) have one solution?

#
# Solution

Transform

$$
\begin{gathered}
\left|\frac{-4 x^{4}-(6 a+10) x^{3}+(16-4 a) x^{2}-\left(6 a^{2}-14 a-40\right) x}{\left(4-x^{2}-a\right)(3 a+2 x+5)}\right|= \\
=\sqrt{a^{2}-2 a+1}
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_1ae08a41583c767d264cg-03.jpg?height=754&width=626&top_left_y=231&top_left_x=1209)

\(\left|\frac{2 x\left(-2 x^{3}-(3 a+5) x^{2}+(8-2 a) x-\left(3 a^{2}-7 a-20\right)\right)}{12 a-3 a x^{2}-3 a^{2}+8 x-2 x^{3}-2 a x+20-5 x^{2}-5 a}\right|=\sqrt{(a-1)^{2}}\)

Solve the equation \(|2 x|=|a-1|, \quad a \neq 4-x^{2}, a \neq-\frac{2}{3} x-\frac{5}{3}, \quad\) graphically in the \(x O a\) system.

That is \(\llbracket_{a=2 x+1}^{a=-2 x+1}, \quad a \neq 4-x^{2}, \quad a \neq-\frac{2}{3} x-\frac{5}{3}\)

Answer: for \(a=-3, a=-1, a=1\).

| Points | Grading Criteria |
| :---: | :--- |
| 20 points | Correct and justified solution |
| 15 points | Functions are correctly transformed, the equation is set up, but a computational error not related to the domain of the function is made in its analytical solution. |
| 10 points | Functions are correctly transformed, the equation is set up, but points not in the domain of the function are not excluded in the graphical solution. |
| 5 points | Expressions are simplified, the equation is set up, but it is not solved or is solved incorrectly. |
| 0 points | Other solutions not meeting the above criteria |