Certainly! Let's address each problem step by step:
### Problem 5: Subtract [tex]\(12 p(3 q - p)\)[/tex] from [tex]\(3 p (6 q + p)\)[/tex]
To subtract [tex]\(12 p (3 q - p)\)[/tex] from [tex]\(3 p (6 q + p)\)[/tex], we follow these steps:
1. Write Down the Expressions:
[tex]\[
\text{Expression 1: } 12 p (3 q - p)
\][/tex]
[tex]\[
\text{Expression 2: } 3 p (6 q + p)
\][/tex]
2. Substitute the Expressions:
We need to subtract Expression 1 from Expression 2.
[tex]\[
(3 p (6 q + p)) - (12 p (3 q - p))
\][/tex]
3. Simplify Each Expression:
Let's first expand both expressions.
For the first expression:
[tex]\[
12 p (3 q - p) = 12 p \cdot 3 q - 12 p \cdot p = 36 p q - 12 p^2
\][/tex]
For the second expression:
[tex]\[
3 p (6 q + p) = 3 p \cdot 6 q + 3 p \cdot p = 18 p q + 3 p^2
\][/tex]
4. Subtract the First Expression from the Second:
[tex]\[
(18 pq + 3 p^2) - (36 pq - 12 p^2)
\][/tex]
Before combining like terms, we'll distribute the subtraction operation:
[tex]\[
18 p q + 3 p^2 - 36 p q + 12 p^2
\][/tex]
5. Combine Like Terms:
[tex]\[
(18 p q - 36 p q) + (3 p^2 + 12 p^2)
\][/tex]
[tex]\[
-18 p q + 15 p^2
\][/tex]
Therefore, the result of subtracting [tex]\(12 p (3 q - p)\)[/tex] from [tex]\(3 p (6 q + p)\)[/tex] is:
[tex]\[
-18 p q + 15 p^2
\][/tex]
### Problem 6: What should be added to [tex]\(-6 p (-p - 1)\)[/tex] to get [tex]\( 3 p^3 + 5 p^2 + 6 p - 2 \)[/tex] ?
To find what should be added to [tex]\(-6 p (-p - 1)\)[/tex] to get [tex]\(3 p^3 + 5 p^2 + 6 p - 2\)[/tex], we proceed as follows:
1. Write Down the Given Expression:
[tex]\[
-6 p (-p - 1)
\][/tex]
2. Simplify the Given Expression:
Expand the given expression:
[tex]\[
-6 p (-p - 1) = -6 p (-p) - (-6 p \cdot 1)
\][/tex]
[tex]\[
= 6 p^2 + 6 p
\][/tex]
3. Set Up the Equation:
We are asked to find some expression [tex]\(x\)[/tex] such that:
[tex]\[
6 p^2 + 6 p + x = 3 p^3 + 5 p^2 + 6 p - 2
\][/tex]
4. Solve For [tex]\(x\)[/tex]:
Subtract [tex]\(6 p^2 + 6 p\)[/tex] from both sides of the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
x = (3 p^3 + 5 p^2 + 6 p - 2) - (6 p^2 + 6 p)
\][/tex]
5. Combine Like Terms:
[tex]\[
x = 3 p^3 + (5 p^2 - 6 p^2) + (6 p - 6 p) - 2
\][/tex]
[tex]\[
x = 3 p^3 - p^2 - 2
\][/tex]
Thus, the expression that should be added to [tex]\(-6 p (-p - 1)\)[/tex] to get [tex]\(3 p^3 + 5 p^2 + 6 p - 2\)[/tex] is:
[tex]\[
3 p^3 - p^2 - 2
\][/tex]
