To multiply [tex]\((a - 3)(b - 4)c\)[/tex] by [tex]\(-6a\)[/tex]:
### Step 1: Understand the expression components.
We are given an expression and a multiplicand:
- Expression: [tex]\((a - 3)(b - 4)c\)[/tex]
- Multiplicand: [tex]\(-6a\)[/tex]
### Step 2: Apply distributive property for multiplication
First, we will multiply the expression [tex]\((a - 3)(b - 4)c\)[/tex] by [tex]\(-6a\)[/tex].
So we have:
[tex]\[
-6a \cdot (a - 3)(b - 4)c
\][/tex]
### Step 3: Start by expanding the inner terms
Expand [tex]\((a - 3)(b - 4)\)[/tex]:
[tex]\[
(a - 3)(b - 4) = ab - 4a - 3b + 12
\][/tex]
Now, include the multiplication by [tex]\(c\)[/tex]:
[tex]\[
(ab - 4a - 3b + 12)c = abc - 4ac - 3bc + 12c
\][/tex]
### Step 4: Multiply the expanded expression by [tex]\(-6a\)[/tex]
Now, we multiply each term in the expanded expression by [tex]\(-6a\)[/tex]:
[tex]\[
-6a \cdot (abc - 4ac - 3bc + 12c)
\][/tex]
Distribute [tex]\(-6a\)[/tex] across all terms:
[tex]\[
-6a \cdot abc + (-6a) \cdot (-4ac) + (-6a) \cdot (-3bc) + (-6a) \cdot (12c)
\][/tex]
Simplify each term:
[tex]\[
-6a \cdot abc = -6a^2bc
\][/tex]
[tex]\[
-6a \cdot (-4ac) = 24a^2c
\][/tex]
[tex]\[
-6a \cdot (-3bc) = 18abc
\][/tex]
[tex]\[
-6a \cdot (12c) = -72ac
\][/tex]
### Step 5: Combine all simplified terms
Combine all the terms to get the final product:
[tex]\[
-6a^2bc + 24a^2c + 18abc - 72ac
\][/tex]
### Conclusion
Thus, the product of [tex]\((a - 3)(b - 4)c\)[/tex] and [tex]\(-6a\)[/tex] simplifies to:
[tex]\[
-6a^2bc + 24a^2c + 18abc - 72ac
\][/tex]
And the expanded product starting from our original multiplication of [tex]\((a - 3)(b - 4)c\)[/tex] by [tex]\(-6a\)[/tex] is:
[tex]\[
-6a(a - 3)(b - 4)c
\][/tex]
