Let's solve the equation step by step:

The equation is:

\[
4(x - 5) = 2(3 - (x + 2)) + 2
\]

### Step 1: Expand both sides.

Expand the left side:

\[
4(x - 5) = 4x - 20
\]

Now, expand the right side. First simplify inside the parentheses:

\[
2(3 - (x + 2)) = 2(3 - x - 2) = 2(1 - x) = 2 - 2x
\]

So the equation becomes:

\[
4x - 20 = 2 - 2x + 2
\]

Simplify the constants on the right side:

\[
4x - 20 = 4 - 2x
\]

### Step 2: Combine like terms.

Move all terms with \(x\) to one side by adding \(2x\) to both sides:

\[
4x + 2x - 20 = 4
\]

Simplify:

\[
6x - 20 = 4
\]

### Step 3: Solve for \(x\).

Add 20 to both sides:

\[
6x = 24
\]

Divide by 6:

\[
x = 4
\]

Thus, the solution is:

\[
x = 4
\]

---

Would you like more details on this solution or have any further questions?

Here are some related questions for further practice:

1. Solve \(3(x - 7) = 5(2 - (x + 1)) + 6\).
2. Solve \(5x + 2 = 3x + 14\).
3. Solve \(2(x + 4) - 3 = 4(x - 1)\).
4. Simplify and solve \(7(2x - 3) = 5(x + 4)\).
5. What happens if you subtract 2x from both sides in \(4x - 20 = 4 - 2x\)?

**Tip:** Always combine like terms and simplify the equation step by step to avoid errors.

This page provides a detailed step-by-step solution to solving the linear equation 4(x-5) = 2(3-(x+2)) + 2, demonstrating key algebraic concepts such as the distributive property and combining like terms. Suitable for students in Grades 7-9.

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