Answer :

We start with the polynomial

[tex]$$
-40a^6b^6 + 25a^2b^5.
$$[/tex]

Step 1. Identify the Common Factors

1. Numerical Factors:
The coefficients are [tex]$-40$[/tex] and [tex]$25$[/tex]. Their greatest common divisor is [tex]$5$[/tex]. Choosing a common factor of [tex]$-5$[/tex] will make the first term positive when it is factored out.

2. Variable Factors:
- For [tex]$a$[/tex]: The smallest power in [tex]$a^6$[/tex] and [tex]$a^2$[/tex] is [tex]$a^2$[/tex].
- For [tex]$b$[/tex]: The smallest power in [tex]$b^6$[/tex] and [tex]$b^5$[/tex] is [tex]$b^5$[/tex].

So, the common factor that can be factored from both terms is

[tex]$$
-5a^2b^5.
$$[/tex]

Step 2. Factor Out the Common Factor

Divide each term by [tex]$-5a^2b^5$[/tex]:

1. For the first term:

[tex]$$
\frac{-40a^6b^6}{-5a^2b^5} = 8a^{6-2}b^{6-5} = 8a^4b.
$$[/tex]

2. For the second term:

[tex]$$
\frac{25a^2b^5}{-5a^2b^5} = -5.
$$[/tex]

Thus, after factoring out the common factor, the polynomial can be written as:

[tex]$$
-5a^2b^5(8a^4b - 5).
$$[/tex]

Final Answer

The complete factored form of the polynomial is:

[tex]$$
-5a^2b^5(8a^4b - 5).
$$[/tex]

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