To find the product of [tex]\((-d + e)(4e + d)\)[/tex], we will use the distributive property, also known as the FOIL method (First, Outside, Inside, Last). This technique helps us expand the expression by multiplying each term in the first binomial by every term in the second binomial.
Let's go through the steps:
1. First: Multiply the first terms of each binomial:
[tex]\[
(-d) \cdot (4e) = -4de
\][/tex]
2. Outside: Multiply the outer terms of the binomials:
[tex]\[
(-d) \cdot (d) = -d^2
\][/tex]
3. Inside: Multiply the inner terms of the binomials:
[tex]\[
(e) \cdot (4e) = 4e^2
\][/tex]
4. Last: Multiply the last terms of each binomial:
[tex]\[
(e) \cdot (d) = ed
\][/tex]
Now, combine the results of these multiplications:
[tex]\[
-4de - d^2 + 4e^2 + ed
\][/tex]
To simplify, notice that the term [tex]\(ed\)[/tex] is the same as [tex]\(de\)[/tex]. When adding these, we get:
[tex]\[
-4de + de - d^2 + 4e^2 = -3de - d^2 + 4e^2
\][/tex]
Thus, the expanded form of the product [tex]\((-d + e)(4e + d)\)[/tex] is:
[tex]\[
-3de - d^2 + 4e^2
\][/tex]
Analyzing the expanded form:
- The number of terms in the product: [tex]\( -3de, -d^2, 4e^2 \)[/tex]. There are 3 terms.
- The degree of each term:
- [tex]\( -3de \)[/tex] has degree 1 (since [tex]\(d^1 \cdot e^1 = de\)[/tex])
- [tex]\( -d^2 \)[/tex] has degree 2
- [tex]\( 4e^2 \)[/tex] has degree 2
The highest degree from these terms is 2.
Therefore, the following statements are true:
- There are 3 terms in the product.
- The product is degree 2.
