Solve the following equations and verify the results.
`4(3x+5)-6(2-x)=8(7+x)+22`

Answer

To solve the equation \( 4(3x + 5) - 6(2 - x) = 8(7 + x) + 22 \) step by step, we will follow the order of operations and isolate the variable \( x \).

### Step 1: Expand the brackets
First, we will distribute the constants outside the brackets.

\[ 4(3x + 5) = 12x + 20 \]
\[ -6(2 - x) = -12 + 6x \]
\[ 8(7 + x) = 56 + 8x \]

Substituting these back into the equation gives:

\[ 12x + 20 - 12 + 6x = 56 + 8x + 22 \]

### Step 2: Simplify both sides
Now, we will simplify both sides of the equation.

On the left side:
\[ 12x + 6x + 20 - 12 = 18x + 8 \]

On the right side:
\[ 56 + 22 = 78 \quad \Rightarrow \quad 78 + 8x = 78 + 8x \]

So the equation now looks like:
\[ 18x + 8 = 78 + 8x \]

### Step 3: Move all variable terms to one side
Next, we will move all terms involving \( x \) to one side and constant terms to the other side.

Subtract \( 8x \) from both sides:
\[ 18x - 8x + 8 = 78 \]
\[ 10x + 8 = 78 \]

### Step 4: Move constant terms to the other side
Now, we will isolate \( 10x \) by subtracting 8 from both sides:
\[ 10x = 78 - 8 \]
\[ 10x = 70 \]

### Step 5: Solve for \( x \)
Finally, divide both sides by 10 to find \( x \):
\[ x = \frac{70}{10} = 7 \]

### Verification
To verify, we will substitute \( x = 7 \) back into the original equation.

Left-hand side (LHS):
\[ 4(3(7) + 5) - 6(2 - 7) = 4(21 + 5) - 6(-5) = 4(26) + 30 = 104 + 30 = 134 \]

Right-hand side (RHS):
\[ 8(7 + 7) + 22 = 8(14) + 22 = 112 + 22 = 134 \]

Since LHS = RHS, our solution is verified.

### Final Answer
The solution to the equation is:
\[ x = 7 \]