4. $\quad$ Transform the expression $\left|\frac{x^{2}-6 x+9}{3-x}+\frac{4 x^{2}-5 x}{x}\right|=\left|\frac{(3-x)^{2}}{3-x}+\frac{x(4 x-5)}{x}\right|=|3 x-2|$, we get $f(x)=|3 x-2|, D(f): x \neq 3, x \neq 0$.

Solve the equation $|3 x-2|=|x+a|$ graphically in the system $x O a$ $(3 x-2)^{2}=(x+a)^{2}$

$$
\begin{gathered}
(3 x-2+x+a)(3 x-2-x-a)=0 \\
{\left[\begin{array}{l}
4 x-2+a=0 \\
2 x-2-a=0
\end{array}\right.} \\
{\left[\begin{array}{c}
a=-4 x+2 \\
a=2 x-2
\end{array}\right.}
\end{gathered}
$$

The equation $f(x)=p(x)$ has exactly one solution if $a=-10$;
Answer: $a=-\frac{2}{3} ; a=2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1fb24413640f25dfaedfg-14.jpg?height=683&width=554&top_left_y=818&top_left_x=1411)

| 15 points | Correct justified solution |
| :--- | :--- |
| 12 points | Functions are correctly transformed, the equation is set up, but a computational error is made in its analytical solution that is not related to the domain of the function. |
| 10 points | Functions are correctly transformed, the equation is set up, but in the graphical solution, points not included in the domain of the function are not excluded. |
| 5 points | Expressions defining the functions are simplified, the equation is set up, but it is not solved or is solved incorrectly. |
| 0 points | Other solutions that do not meet the criteria listed above |