Let's solve the equation step by step:

Given equation:

\[
2(5c + 2) - 2c = 3(2c + 3) + 7
\]

### Step 1: Expand both sides.
On the left-hand side:
\[
2(5c + 2) = 10c + 4
\]
So:
\[
10c + 4 - 2c = 3(2c + 3) + 7
\]

On the right-hand side:
\[
3(2c + 3) = 6c + 9
\]
So:
\[
10c + 4 - 2c = 6c + 9 + 7
\]

### Step 2: Simplify both sides.
Left-hand side:
\[
10c - 2c + 4 = 8c + 4
\]

Right-hand side:
\[
6c + 9 + 7 = 6c + 16
\]

Now the equation becomes:
\[
8c + 4 = 6c + 16
\]

### Step 3: Move all the terms involving \( c \) to one side.
Subtract \( 6c \) from both sides:
\[
8c - 6c + 4 = 16
\]
\[
2c + 4 = 16
\]

### Step 4: Solve for \( c \).
Subtract 4 from both sides:
\[
2c = 16 - 4
\]
\[
2c = 12
\]

Now, divide by 2:
\[
c = \frac{12}{2} = 6
\]

### Final Answer:
\[
c = 6
\]

Would you like any further details on the solution or any clarifications?

Here are some related questions for practice:
1. Solve \( 3(4x - 1) + 2x = 5(x + 2) \).
2. Solve \( 7(2d + 1) - 4d = 5(3d + 4) \).
3. Solve \( 4(3y - 2) - 6y = 2(5y + 3) + 4 \).
4. Solve \( 5(2b + 3) + 6b = 7(b + 4) + 9 \).
5. Solve \( 6(a + 4) - 3a = 2(3a + 5) + 8 \).

**Tip:** Always expand parentheses first and then simplify both sides before solving for the variable.

Step-by-step solution to the linear equation 2(5c+2)-2c=3(2c+3)+7, using the distributive property and simplification techniques. This problem is ideal for students in Grades 6-8 and focuses on key algebraic concepts.

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