In a product pyramid, each field (except those on the bottom floor) contains the product of the expressions written in the two fields directly below it. Fill in the missing expressions in the fields of the product pyramid in the image. Try to write the expressions in their simplest form and indicate the order in which you filled them in $(x \neq 0)$.

( $S$. Bednářová)

![](https://cdn.mathpix.com/cropped/2024_04_17_ac63ecbffe7e6b0bc0e7g-1.jpg?height=254&width=771&top_left_y=747&top_left_x=677)
First, we will complete the 3rd expression in the bottom row. It will be $3 x: \frac{1}{x^{2}}=\underline{\underline{3 x^{3}}}$

The last expression in the 2nd row from the bottom is $3 x^{3} \cdot \frac{x+2}{3 x^{3}}=\underline{\underline{x+2}}$.

The last expression in the 3rd row from the bottom is $3 x \cdot(x+2)=\underline{\underline{3 x^{2}+6 x}}$. (Any of the underlined expressions can be filled into the pyramid.)

The first expression in the 3rd row from the bottom is $\frac{9 x^{4}-36 x^{2}}{3 x^{2}+6 x}=\frac{\left(3 x^{2}+6 x\right)\left(3 x^{2}-6 x\right)}{3 x^{2}+6 x}=\underline{\underline{3 x^{2}-6 x}}=$ $=3 x \cdot(x-2)$.

The first expression in the 2nd row from the bottom is $\frac{3 x^{2}-6 x}{3 x}=\frac{3 x(x-2)}{3 x}=\underline{\underline{x-2}}$.

The first expression in the 1st row from the bottom is $(x-2): \frac{1}{x^{2}}=\underline{\underline{(x-2) \cdot x^{2}}}=\underline{\underline{x^{3}-2 x^{2}}}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_ac63ecbffe7e6b0bc0e7g-1.jpg?height=248&width=771&top_left_y=1738&top_left_x=677)

(1 point is awarded for correctly filling in each number.)

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