To find the product of the given expression and then simplify it, we start with the expression:
[tex]\[
-4a \left( -9a^4 - 9a - 9 \right)
\][/tex]
We will distribute [tex]\(-4a\)[/tex] across each term inside the parentheses:
1. Distributing [tex]\(-4a\)[/tex] to [tex]\(-9a^4\)[/tex]:
[tex]\[
-4a \cdot -9a^4 = 36a \cdot a^4 = 36a^5
\][/tex]
2. Distributing [tex]\(-4a\)[/tex] to [tex]\(-9a\)[/tex]:
[tex]\[
-4a \cdot -9a = 36a \cdot a = 36a^2
\][/tex]
3. Distributing [tex]\(-4a\)[/tex] to [tex]\(-9\)[/tex]:
[tex]\[
-4a \cdot -9 = 36a
\][/tex]
Combining all the distributed terms together, we get:
[tex]\[
36a^5 + 36a^2 + 36a
\][/tex]
Now, we can factor out the common factor [tex]\(36a\)[/tex] from each term in the expression:
[tex]\[
36a(a^4 + a + 1)
\][/tex]
Therefore, the simplified form of the product [tex]\(-4a(-9a^4 - 9a - 9)\)[/tex] is:
[tex]\[
\boxed{36a(a^4 + a + 1)}
\][/tex]
