To expand and express [tex]\(\left(-p^2 + 4p - 3\right)\left(p^2 + 2\right)\)[/tex] as a polynomial in standard form, follow these steps:
1. Distribute each term in the first polynomial [tex]\((-p^2 + 4p - 3)\)[/tex] through the second polynomial [tex]\((p^2 + 2)\)[/tex].
Let's distribute each term individually:
- Distribute [tex]\(-p^2\)[/tex] through [tex]\((p^2 + 2)\)[/tex]:
[tex]\[
-p^2(p^2 + 2) = -p^2 \cdot p^2 + (-p^2 \cdot 2) = -p^4 - 2p^2
\][/tex]
- Distribute [tex]\(4p\)[/tex] through [tex]\((p^2 + 2)\)[/tex]:
[tex]\[
4p(p^2 + 2) = 4p \cdot p^2 + 4p \cdot 2 = 4p^3 + 8p
\][/tex]
- Distribute [tex]\(-3\)[/tex] through [tex]\((p^2 + 2)\)[/tex]:
[tex]\[
-3(p^2 + 2) = -3 \cdot p^2 + (-3 \cdot 2) = -3p^2 - 6
\][/tex]
2. Combine all the distributed parts:
[tex]\[
-p^4 - 2p^2 + 4p^3 + 8p - 3p^2 - 6
\][/tex]
3. Combine like terms to simplify the expression:
[tex]\[
-p^4 + 4p^3 - 5p^2 + 8p - 6
\][/tex]
Thus, the polynomial in standard form is:
[tex]\[
\left(-p^2 + 4p - 3\right)\left(p^2 + 2\right) = -p^4 + 4p^3 - 5p^2 + 8p - 6
\][/tex]
