subtract 3a(a+b+c)-2ab(a-b+c)from 4c(-a+b+c)
Multiply them out, perform the subtraction, and collect the terms. There will be two terms with the combination ac (-4ac -3ac = -7ac); all the others will have unique combinations of a, b and c.
To subtract the expression 3a(a+b+c) - 2ab(a-b+c) from 4c(-a+b+c), you need to distribute the multiplication and then combine like terms.
Let's break it down step by step:
Step 1: Distribute the multiplication in both expressions.
In the expression 3a(a+b+c), distribute the multiplication to each term inside the parentheses:
3a(a+b+c) = 3a^2 + 3ab + 3ac
In the expression -2ab(a-b+c), distribute the multiplication to each term inside the parentheses:
-2ab(a-b+c) = -2a^2b + 2ab^2 - 2abc
In the expression 4c(-a+b+c), distribute the multiplication to each term inside the parentheses:
4c(-a+b+c) = -4ac + 4bc + 4c^2
Now, let's rewrite the expression with the distributed terms:
4c(-a+b+c) = -4ac + 4bc + 4c^2
Step 2: Combine like terms.
Now we can subtract the expression 3a(a+b+c) - 2ab(a-b+c) from 4c(-a+b+c) by subtracting like terms:
-4ac + 4bc + 4c^2 - (3a^2 + 3ab + 3ac - 2a^2b + 2ab^2 - 2abc)
Step 3: Simplify the expression by rearranging the terms.
Combine like terms:
-4ac + 4bc + 4c^2 - 3a^2 - 3ab - 3ac + 2a^2b - 2ab^2 + 2abc
Step 4: Rearrange the terms to simplify further.
-3a^2 + 2a^2b - 3ab - 3ac + 2abc - 2ab^2 - 4ac + 4bc + 4c^2
So, the simplified expression after subtracting 3a(a+b+c) - 2ab(a-b+c) from 4c(-a+b+c) is:
-3a^2 + 2a^2b - 3ab - 3ac + 2abc - 2ab^2 - 4ac + 4bc + 4c^2