Subtract `4(-ac+4bc+c^(2))` from `3(a^(2)+ab+ac)-5(ab-b^(2)+bc)`.

Answer

To solve the problem of subtracting \( 4(-ac + 4bc + c^2) \) from \( 3(a^2 + ab + ac) - 5(ab - b^2 + bc) \), we will follow these steps:

### Step 1: Simplify the expression \( 3(a^2 + ab + ac) \)

First, we distribute the 3 across the terms inside the parentheses:

\[ 3(a^2 + ab + ac) = 3a^2 + 3ab + 3ac \]

**Hint:** When distributing, multiply each term inside the parentheses by the number outside.

### Step 2: Simplify the expression \( -5(ab - b^2 + bc) \)

Next, we distribute the -5 across the terms inside the parentheses:

\[ -5(ab - b^2 + bc) = -5ab + 5b^2 - 5bc \]

**Hint:** Remember that distributing a negative number changes the signs of the terms inside the parentheses.

### Step 3: Combine the results from Steps 1 and 2

Now we combine the two simplified expressions:

\[ 3a^2 + 3ab + 3ac - 5ab + 5b^2 - 5bc \]

Combine like terms:

- For \( ab \): \( 3ab - 5ab = -2ab \)
- The other terms remain the same.

So we have:

\[ 3a^2 - 2ab + 3ac + 5b^2 - 5bc \]

**Hint:** Group similar terms together to simplify the expression.

### Step 4: Rewrite the expression to be subtracted

Now we need to rewrite the expression \( 4(-ac + 4bc + c^2) \):

Distributing the 4 gives:

\[ 4(-ac + 4bc + c^2) = -4ac + 16bc + 4c^2 \]

**Hint:** Again, remember to distribute carefully, paying attention to the signs.

### Step 5: Subtract the second expression from the first

Now we subtract \( -4ac + 16bc + 4c^2 \) from \( 3a^2 - 2ab + 3ac + 5b^2 - 5bc \):

\[ (3a^2 - 2ab + 3ac + 5b^2 - 5bc) - (-4ac + 16bc + 4c^2) \]

This becomes:

\[ 3a^2 - 2ab + 3ac + 5b^2 - 5bc + 4ac - 16bc - 4c^2 \]

### Step 6: Combine like terms again

Now we combine the like terms:

- For \( ac \): \( 3ac + 4ac = 7ac \)
- For \( bc \): \( -5bc - 16bc = -21bc \)
- The other terms remain the same.

So we have:

\[ 3a^2 - 2ab + 7ac + 5b^2 - 21bc - 4c^2 \]

### Final Answer

Thus, the final result after performing the subtraction is:

\[ 3a^2 - 2ab + 7ac + 5b^2 - 21bc - 4c^2 \]

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