Multiplying Polynomials Practice Worksheet with ten algebraic expressions to solve.

The task involves multiplying polynomials. Below, I will solve each problem step by step and explain the process.

---

Problem 1:

\[
(p + 2)(p - 10)
\]

Solution:
Use the distributive property (also known as the FOIL method for binomials):
\[
(p + 2)(p - 10) = p \cdot p + p \cdot (-10) + 2 \cdot p + 2 \cdot (-10)
\]
\[
= p^2 - 10p + 2p - 20
\]
Combine like terms:
\[
= p^2 - 8p - 20
\]

Answer:
\[
\boxed{p^2 - 8p - 20}
\]

---

Problem 2:

\[
(x - 8)(x + 9)
\]

Solution:
Again, use the distributive property:
\[
(x - 8)(x + 9) = x \cdot x + x \cdot 9 + (-8) \cdot x + (-8) \cdot 9
\]
\[
= x^2 + 9x - 8x - 72
\]
Combine like terms:
\[
= x^2 + x - 72
\]

Answer:
\[
\boxed{x^2 + x - 72}
\]

---

Problem 3:

\[
(u + 5)(u - 5)
\]

Solution:
This is a difference of squares, which follows the pattern \((a + b)(a - b) = a^2 - b^2\):
\[
(u + 5)(u - 5) = u^2 - 5^2
\]
\[
= u^2 - 25
\]

Answer:
\[
\boxed{u^2 - 25}
\]

---

Problem 4:

\[
(b + 2)(b + 11)
\]

Solution:
Use the distributive property:
\[
(b + 2)(b + 11) = b \cdot b + b \cdot 11 + 2 \cdot b + 2 \cdot 11
\]
\[
= b^2 + 11b + 2b + 22
\]
Combine like terms:
\[
= b^2 + 13b + 22
\]

Answer:
\[
\boxed{b^2 + 13b + 22}
\]

---

Problem 5:

\[
(3b + 5)(b - 1)
\]

Solution:
Use the distributive property:
\[
(3b + 5)(b - 1) = 3b \cdot b + 3b \cdot (-1) + 5 \cdot b + 5 \cdot (-1)
\]
\[
= 3b^2 - 3b + 5b - 5
\]
Combine like terms:
\[
= 3b^2 + 2b - 5
\]

Answer:
\[
\boxed{3b^2 + 2b - 5}
\]

---

Problem 6:

\[
(4a - 5)^2
\]

Solution:
This is a square of a binomial, which follows the pattern \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(4a - 5)^2 = (4a)^2 - 2(4a)(5) + 5^2
\]
\[
= 16a^2 - 40a + 25
\]

Answer:
\[
\boxed{16a^2 - 40a + 25}
\]

---

Problem 7:

\[
(3r - 4)^2
\]

Solution:
This is another square of a binomial, following the pattern \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(3r - 4)^2 = (3r)^2 - 2(3r)(4) + 4^2
\]
\[
= 9r^2 - 24r + 16
\]

Answer:
\[
\boxed{9r^2 - 24r + 16}
\]

---

Problem 8:

\[
(y + 3)(9y - 2)
\]

Solution:
Use the distributive property:
\[
(y + 3)(9y - 2) = y \cdot 9y + y \cdot (-2) + 3 \cdot 9y + 3 \cdot (-2)
\]
\[
= 9y^2 - 2y + 27y - 6
\]
Combine like terms:
\[
= 9y^2 + 25y - 6
\]

Answer:
\[
\boxed{9y^2 + 25y - 6}
\]

---

Problem 9:

\[
4a^2(a^2 + 5a - 1)
\]

Solution:
Distribute \(4a^2\) to each term inside the parentheses:
\[
4a^2(a^2 + 5a - 1) = 4a^2 \cdot a^2 + 4a^2 \cdot 5a + 4a^2 \cdot (-1)
\]
\[
= 4a^4 + 20a^3 - 4a^2
\]

Answer:
\[
\boxed{4a^4 + 20a^3 - 4a^2}
\]

---

Problem 10:

\[
x^2(2x^2 + 3x - 4)
\]

Solution:
Distribute \(x^2\) to each term inside the parentheses:
\[
x^2(2x^2 + 3x - 4) = x^2 \cdot 2x^2 + x^2 \cdot 3x + x^2 \cdot (-4)
\]
\[
= 2x^4 + 3x^3 - 4x^2
\]

Answer:
\[
\boxed{2x^4 + 3x^3 - 4x^2}
\]

---

Final Answers:

1. \(\boxed{p^2 - 8p - 20}\)
2. \(\boxed{x^2 + x - 72}\)
3. \(\boxed{u^2 - 25}\)
4. \(\boxed{b^2 + 13b + 22}\)
5. \(\boxed{3b^2 + 2b - 5}\)
6. \(\boxed{16a^2 - 40a + 25}\)
7. \(\boxed{9r^2 - 24r + 16}\)
8. \(\boxed{9y^2 + 25y - 6}\)
9. \(\boxed{4a^4 + 20a^3 - 4a^2}\)
10. \(\boxed{2x^4 + 3x^3 - 4x^2}\)

---

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
1. & \ p^2 - 8p - 20 \\
2. & \ x^2 + x - 72 \\
3. & \ u^2 - 25 \\
4. & \ b^2 + 13b + 22 \\
5. & \ 3b^2 + 2b - 5 \\
6. & \ 16a^2 - 40a + 25 \\
7. & \ 9r^2 - 24r + 16 \\
8. & \ 9y^2 + 25y - 6 \\
9. & \ 4a^4 + 20a^3 - 4a^2 \\
10. & \ 2x^4 + 3x^3 - 4x^2
\end{aligned}
}
\]