To find the product of [tex]\((-d + e)(4e + d)\)[/tex], we need to distribute the terms and then simplify the expression. Let's go through the steps carefully:
### Step-by-step Solution:
1. Distribute the terms:
[tex]\[
(-d + e)(4e + d) = (-d)(4e) + (-d)(d) + (e)(4e) + (e)(d)
\][/tex]
2. Perform each multiplication:
[tex]\[
(-d)(4e) = -4de
\][/tex]
[tex]\[
(-d)(d) = -d^2
\][/tex]
[tex]\[
(e)(4e) = 4e^2
\][/tex]
[tex]\[
(e)(d) = de
\][/tex]
3. Combine all the terms:
[tex]\[
(-d)(4e) + (-d)(d) + (e)(4e) + (e)(d)
= -4de - d^2 + 4e^2 + de
\][/tex]
4. Combine like terms:
[tex]\[
-4de + de = -3de
\][/tex]
So, the expression simplifies to:
[tex]\[
-d^2 - 3de + 4e^2
\][/tex]
### Evaluating the Statements:
1. Number of terms in the product:
- The simplified expression [tex]\(-d^2 - 3de + 4e^2\)[/tex] has 3 terms: [tex]\(-d^2\)[/tex], [tex]\(-3de\)[/tex], and [tex]\(4e^2\)[/tex].
- Therefore, the statement "There are 2 terms in the product." is false.
- The statement "There are 3 terms in the product." is true.
- The statement "There are 4 terms in the product." is false.
2. Degree of the product:
- The degree of a polynomial is the highest degree of any term in the polynomial.
- [tex]\(-d^2\)[/tex] has degree 2 (since [tex]\(d^2\)[/tex] is a term).
- [tex]\(-3de\)[/tex] has degree 2 (since [tex]\(d\)[/tex] and [tex]\(e\)[/tex] are both first degree, and [tex]\(1+1=2\)[/tex]).
- [tex]\(4e^2\)[/tex] has degree 2 (since [tex]\(e^2\)[/tex] is a term).
- The highest degree among these terms is 2.
- Therefore, the statement "The product is degree 1." is false.
- The statement "The product is degree 2." is true.
- The statement "The product is degree 4." is false.
### Summary of True Statements:
- There are 3 terms in the product.
- The product is degree 2.
So, the final answer to which statements are true is:
- There are 3 terms in the product.
- The product is degree 2.
