Handwritten math worksheet featuring 13 distributive property problems for simplification.

Problem: Simplify the given expressions using the distributive property.

The distributive property states:
\[
a(b + c) = ab + ac
\]
We will apply this property to each expression step by step.

---

Expression 1: \( 8(x + 4) \)

Using the distributive property:
\[
8(x + 4) = 8 \cdot x + 8 \cdot 4 = 8x + 32
\]

Simplified: \( 8x + 32 \)

---

Expression 2: \( (4a - 1)2 \)

Rewrite as:
\[
(4a - 1)2 = 2(4a - 1)
\]
Using the distributive property:
\[
2(4a - 1) = 2 \cdot 4a + 2 \cdot (-1) = 8a - 2
\]

Simplified: \( 8a - 2 \)

---

Expression 3: \( -2(3x - 5) \)

Using the distributive property:
\[
-2(3x - 5) = -2 \cdot 3x + (-2) \cdot (-5) = -6x + 10
\]

Simplified: \( -6x + 10 \)

---

Expression 4: \( 2(3x - 1) + x \)

First, distribute \( 2 \) in \( 2(3x - 1) \):
\[
2(3x - 1) = 2 \cdot 3x + 2 \cdot (-1) = 6x - 2
\]
Now add \( x \):
\[
6x - 2 + x = 7x - 2
\]

Simplified: \( 7x - 2 \)

---

Expression 5: \( 6r + 2(r + 4) \)

First, distribute \( 2 \) in \( 2(r + 4) \):
\[
2(r + 4) = 2 \cdot r + 2 \cdot 4 = 2r + 8
\]
Now add \( 6r \):
\[
6r + 2r + 8 = 8r + 8
\]

Simplified: \( 8r + 8 \)

---

Expression 6: \( 12y - (y - 7) \)

Distribute the negative sign in \( -(y - 7) \):
\[
-(y - 7) = -y + 7
\]
Now combine terms:
\[
12y - y + 7 = 11y + 7
\]

Simplified: \( 11y + 7 \)

---

Expression 7: \( 3(m + 5) - 10 \)

First, distribute \( 3 \) in \( 3(m + 5) \):
\[
3(m + 5) = 3 \cdot m + 3 \cdot 5 = 3m + 15
\]
Now subtract 10:
\[
3m + 15 - 10 = 3m + 5
\]

Simplified: \( 3m + 5 \)

---

Expression 8: \( -4(x + 2) + 2x \)

First, distribute \( -4 \) in \( -4(x + 2) \):
\[
-4(x + 2) = -4 \cdot x + (-4) \cdot 2 = -4x - 8
\]
Now add \( 2x \):
\[
-4x - 8 + 2x = -2x - 8
\]

Simplified: \( -2x - 8 \)

---

Expression 9: \( 7(w - 5) + 3w \)

First, distribute \( 7 \) in \( 7(w - 5) \):
\[
7(w - 5) = 7 \cdot w + 7 \cdot (-5) = 7w - 35
\]
Now add \( 3w \):
\[
7w - 35 + 3w = 10w - 35
\]

Simplified: \( 10w - 35 \)

---

Expression 10: \( 5(x + 2) - 3(x - 4) \)

First, distribute \( 5 \) in \( 5(x + 2) \):
\[
5(x + 2) = 5 \cdot x + 5 \cdot 2 = 5x + 10
\]
Next, distribute \( -3 \) in \( -3(x - 4) \):
\[
-3(x - 4) = -3 \cdot x + (-3) \cdot (-4) = -3x + 12
\]
Now combine terms:
\[
5x + 10 - 3x + 12 = (5x - 3x) + (10 + 12) = 2x + 22
\]

Simplified: \( 2x + 22 \)

---

Expression 11: \( -2(c - 2) + 4(c - 1) \)

First, distribute \( -2 \) in \( -2(c - 2) \):
\[
-2(c - 2) = -2 \cdot c + (-2) \cdot (-2) = -2c + 4
\]
Next, distribute \( 4 \) in \( 4(c - 1) \):
\[
4(c - 1) = 4 \cdot c + 4 \cdot (-1) = 4c - 4
\]
Now combine terms:
\[
-2c + 4 + 4c - 4 = (-2c + 4c) + (4 - 4) = 2c + 0 = 2c
\]

Simplified: \( 2c \)

---

Expression 12: \( 3(b + 5) - (b - 2) \)

First, distribute \( 3 \) in \( 3(b + 5) \):
\[
3(b + 5) = 3 \cdot b + 3 \cdot 5 = 3b + 15
\]
Next, distribute the negative sign in \( -(b - 2) \):
\[
-(b - 2) = -b + 2
\]
Now combine terms:
\[
3b + 15 - b + 2 = (3b - b) + (15 + 2) = 2b + 17
\]

Simplified: \( 2b + 17 \)

---

Expression 13: \( 4(x - 2) + 3(x + 9) \)

First, distribute \( 4 \) in \( 4(x - 2) \):
\[
4(x - 2) = 4 \cdot x + 4 \cdot (-2) = 4x - 8
\]
Next, distribute \( 3 \) in \( 3(x + 9) \):
\[
3(x + 9) = 3 \cdot x + 3 \cdot 9 = 3x + 27
\]
Now combine terms:
\[
4x - 8 + 3x + 27 = (4x + 3x) + (-8 + 27) = 7x + 19
\]

Simplified: \( 7x + 19 \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 8x + 32 \\
2. & \ 8a - 2 \\
3. & \ -6x + 10 \\
4. & \ 7x - 2 \\
5. & \ 8r + 8 \\
6. & \ 11y + 7 \\
7. & \ 3m + 5 \\
8. & \ -2x - 8 \\
9. & \ 10w - 35 \\
10. & \ 2x + 22 \\
11. & \ 2c \\
12. & \ 2b + 17 \\
13. & \ 7x + 19
\end{aligned}
}
\]