Question

Multiply. Write your answer in simplest form.
3h–5
5h+1
h+2
20h2–8h–1

Answer 1

To multiply the given expressions \( (3h - 5)(5h + 1)(h + 2)(20h^2 - 8h - 1) \), we can follow these steps:

1. **Multiply the first two expressions:**
\[
(3h - 5)(5h + 1) = 3h \cdot 5h + 3h \cdot 1 - 5 \cdot 5h - 5 \cdot 1
\]
\[
= 15h^2 + 3h - 25h - 5 = 15h^2 - 22h - 5
\]

2. **Multiply this result with the third expression:**
\[
(15h^2 - 22h - 5)(h + 2) = 15h^2 \cdot h + 15h^2 \cdot 2 - 22h \cdot h - 22h \cdot 2 - 5 \cdot h - 5 \cdot 2
\]
\[
= 15h^3 + 30h^2 - 22h^2 - 44h - 5h - 10
\]
\[
= 15h^3 + (30h^2 - 22h^2) + (-44h - 5h) - 10
\]
\[
= 15h^3 + 8h^2 - 49h - 10
\]

3. **Finally, multiply this with the fourth expression:**
\[
(15h^3 + 8h^2 - 49h - 10)(20h^2 - 8h - 1)
\]
We'll break this down further and multiply term by term.

- \( 15h^3 \) with \( 20h^2 \):
\[ 15h^3 \cdot 20h^2 = 300h^5 \]
- \( 15h^3 \) with \( -8h \):
\[ 15h^3 \cdot -8h = -120h^4 \]
- \( 15h^3 \) with \( -1 \):
\[ 15h^3 \cdot -1 = -15h^3 \]
- \( 8h^2 \) with \( 20h^2 \):
\[ 8h^2 \cdot 20h^2 = 160h^4 \]
- \( 8h^2 \) with \( -8h \):
\[ 8h^2 \cdot -8h = -64h^3 \]
- \( 8h^2 \) with \( -1 \):
\[ 8h^2 \cdot -1 = -8h^2 \]
- \( -49h \) with \( 20h^2 \):
\[ -49h \cdot 20h^2 = -980h^3 \]
- \( -49h \) with \( -8h \):
\[ -49h \cdot -8h = 392h^2 \]
- \( -49h \) with \( -1 \):
\[ -49h \cdot -1 = 49h \]
- \( -10 \) with \( 20h^2 \):
\[ -10 \cdot 20h^2 = -200h^2 \]
- \( -10 \) with \( -8h \):
\[ -10 \cdot -8h = 80h \]
- \( -10 \) with \( -1 \):
\[ -10 \cdot -1 = 10 \]

4. **Now combine all the terms:**
\[
300h^5 + (-120h^4 + 160h^4) + (-15h^3 - 64h^3 - 980h^3) + (-8h^2 + 392h^2 - 200h^2) + (49h + 80h) + 10
\]
\[
= 300h^5 + 40h^4 + (-15 - 64 - 980)h^3 + (392 - 200 - 8)h^2 + (49 + 80)h + 10
\]
\[
= 300h^5 + 40h^4 - 1059h^3 + 184h^2 + 129h + 10
\]

So the final answer in simplest form is:
\[
\boxed{300h^5 + 40h^4 - 1059h^3 + 184h^2 + 129h + 10}
\]