To find the product [tex]\((p+5)(p-2)\)[/tex], we can use the distributive property of multiplication over addition, often called the FOIL method (First, Outer, Inner, Last). Let's go through this step-by-step:
1. First: Multiply the first terms of each binomial.
[tex]\[
p \cdot p = p^2
\][/tex]
2. Outer: Multiply the outer terms of each binomial.
[tex]\[
p \cdot (-2) = -2p
\][/tex]
3. Inner: Multiply the inner terms of each binomial.
[tex]\[
5 \cdot p = 5p
\][/tex]
4. Last: Multiply the last terms of each binomial.
[tex]\[
5 \cdot (-2) = -10
\][/tex]
Now, add all these products together:
[tex]\[
p^2 - 2p + 5p - 10
\][/tex]
Combine the like terms ([tex]\(-2p\)[/tex] and [tex]\(5p\)[/tex]):
[tex]\[
p^2 + (5p - 2p) - 10
\][/tex]
[tex]\[
p^2 + 3p - 10
\][/tex]
Hence, the expression [tex]\((p+5)(p-2)\)[/tex] expands to [tex]\(p^2 + 3p - 10\)[/tex].
Comparing this with the given options:
- A: [tex]\(p^2 + 3p - 10\)[/tex]
- B: [tex]\(p^2 - 10\)[/tex]
- C: [tex]\(p^2 + 7p - 10\)[/tex]
- D: [tex]\(p^2 - 3p\)[/tex]
The correct answer is option A: [tex]\(p^2 + 3p - 10\)[/tex].
