Answer :

We start with the expression

[tex]$$
45x^2 - 90x^3 + 9x^2y^2.
$$[/tex]

Step 1. Identify the Common Factor

Look at each term:

- The first term is [tex]$45x^2$[/tex].
- The second term is [tex]$-90x^3$[/tex].
- The third term is [tex]$9x^2y^2$[/tex].

Notice that each term has a factor of [tex]$9x^2$[/tex]. We can factor [tex]$9x^2$[/tex] out of each term.

Step 2. Factor Out [tex]$9x^2$[/tex]

Divide each term by [tex]$9x^2$[/tex]:

- [tex]$45x^2$[/tex] divided by [tex]$9x^2$[/tex] gives [tex]$5$[/tex], because [tex]$\frac{45}{9} = 5$[/tex].
- [tex]$-90x^3$[/tex] divided by [tex]$9x^2$[/tex] gives [tex]$-10x$[/tex], since [tex]$\frac{-90}{9} = -10$[/tex] and [tex]$x^3/x^2 = x$[/tex].
- [tex]$9x^2y^2$[/tex] divided by [tex]$9x^2$[/tex] gives [tex]$y^2$[/tex], because the [tex]$9x^2$[/tex] cancel.

So, the expression becomes

[tex]$$
45x^2 - 90x^3 + 9x^2y^2 = 9x^2(5 - 10x + y^2).
$$[/tex]

Step 3. Write the Final Factored Form

We can rearrange the factors inside the parentheses for clarity. Reordering the terms inside the parentheses, one acceptable form is

[tex]$$
9x^2(-10x + y^2 + 5).
$$[/tex]

Thus, the fully factored expression is

[tex]$$
\boxed{9x^2(-10x + y^2 + 5)}.
$$[/tex]

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