Let's carefully analyze and solve the given problem step-by-step.
Given expressions are [tex]\(3x\)[/tex], [tex]\(-2x\)[/tex], [tex]\(y\)[/tex], and [tex]\(-3y\)[/tex].
### a) Like terms
Like terms are terms that have the same variable raised to the same power.
- [tex]\(3x\)[/tex] and [tex]\(-2x\)[/tex] are like terms because both have the variable [tex]\(x\)[/tex].
- [tex]\(y\)[/tex] and [tex]\(-3y\)[/tex] are like terms because both have the variable [tex]\(y\)[/tex].
So, the like terms are:
[tex]\[ (3x, -2x) \][/tex]
[tex]\[ (y, -3y) \][/tex]
### b) Product of these expressions
To find the product of these expressions, we multiply all the coefficients together and write the variables with their respective powers.
- Coefficients: [tex]\(3\)[/tex], [tex]\(-2\)[/tex], [tex]\(1\)[/tex] (coefficient of [tex]\(y\)[/tex]), and [tex]\(-3\)[/tex]
Let's multiply the coefficients:
[tex]\[ 3 \times (-2) \times 1 \times (-3) = 18 \][/tex]
Now, let's consider the variables:
[tex]\[ x \times x = x^2 \][/tex]
[tex]\[ y \times y^{-1} \times y = y^{-1} y = y^0 = 1 \][/tex]
So, the product of these expressions is:
[tex]\[ 18 \][/tex]
### c) Sum of these expressions and multiply the sum by [tex]\((x + 2y)\)[/tex]
First, let's find the sum of the given expressions:
[tex]\[
3x + (-2x) + y + (-3y)
\][/tex]
Combine the like terms:
[tex]\[ 3x - 2x = x \][/tex]
[tex]\[ y - 3y = -2y \][/tex]
Thus, the sum of the expressions is:
[tex]\[ x - 2y \][/tex]
Next, we need to multiply this sum by [tex]\((x + 2y)\)[/tex]:
[tex]\[
(x - 2y)(x + 2y)
\][/tex]
We use the distributive property (FOIL method):
[tex]\[
x(x) + x(2y) - 2y(x) - 2y(2y)
\][/tex]
Simplify each term:
[tex]\[ x^2 + 2xy - 2xy - 4y^2 \][/tex]
Combine like terms:
[tex]\[ x^2 + 0 - 4y^2 = x^2 - 4y^2 \][/tex]
Therefore, the result of multiplying the sum by [tex]\((x + 2y)\)[/tex] is:
[tex]\[ x^2 - 4y^2 \][/tex]
