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Multiply the following:

[tex]\[ (a - 3b - 4c) \cdot (-6a) \][/tex]


Sagot :

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]