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Sagot :
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]
[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]
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