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:

$x = 4$

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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.