9. For what values of the parameter $\boldsymbol{a}$ does the equation $|f(x)-4|=p(x)$, where $f(x)=\left|\frac{x^{2}+3 x}{x+3}-\frac{x^{2}-4 x+4}{2-x}\right|$, $p(x)=a$ have three solutions? If there is more than one value of the parameter, indicate their product in the answer.
# Solution:

Simplify $f(x)=\left|\frac{x^{2}+3 x}{x+3}-\frac{x^{2}-4 x+4}{2-x}\right|$, we get $f(x)=|2 x-2|$, where $x \neq-3, x \neq 2$.

Solve the equation || $2 x-2|-4|=a$, where $x \neq-3, x \neq 2$ graphically in the system $x O a$.

The equation has three solutions when $a=2$.

The product is 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_de39ee82fbf77ee5c736g-3.jpg?height=494&width=651&top_left_y=1969&top_left_x=1274)

## Answer: 2.

## Solution for Variant #2