Which pair shows equivalent expressions? $$ 2 x+10=-2(x-5) $$ $$ -2(x+5)=2 x-10 $$ $$ -2 x-10=-2(x+5) $$ $$ -2(x-5)=-2 x-10 $$

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Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$-2\left(x-5\right)=-2x-10$$
- step1: Expand the expression:
  $$-2x+10=-2x-10$$
- step2: Cancel equal terms:
  $$10=-10$$
- step3: The statement is false:
  $$x \in \varnothing $$
Solve the equation $$ -2(x+5)=2 x-10 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$-2\left(x+5\right)=2x-10$$
- step1: Expand the expression:
  $$-2x-10=2x-10$$
- step2: Cancel equal terms:
  $$-2x=2x$$
- step3: Add or subtract both sides:
  $$-2x-2x=0$$
- step4: Subtract the terms:
  $$-4x=0$$
- step5: Change the signs:
  $$4x=0$$
- step6: Rewrite the expression:
  $$x=0$$
Solve the equation $$ -2 x-10=-2(x+5) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$-2x-10=-2\left(x+5\right)$$
- step1: Expand the expression:
  $$-2x-10=-2x-10$$
- step2: The statement is true:
  $$x \in \mathbb{R}$$
Solve the equation $$ 2 x+10=-2(x-5) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2x+10=-2\left(x-5\right)$$
- step1: Expand the expression:
  $$2x+10=-2x+10$$
- step2: Cancel equal terms:
  $$2x=-2x$$
- step3: Add or subtract both sides:
  $$2x-\left(-2x\right)=0$$
- step4: Subtract the terms:
  $$4x=0$$
- step5: Rewrite the expression:
  $$x=0$$
Let's analyze the given pairs of equations to determine which ones show equivalent expressions by solving each equation for $$ x $$.

1. **First equation:** 
   $$
   2x + 10 = -2(x - 5)
   $$
   Solving this gives:
   $$
   x = 0
   $$

2. **Second equation:** 
   $$
   -2(x + 5) = 2x - 10
   $$
   Solving this also gives:
   $$
   x = 0
   $$

3. **Third equation:** 
   $$
   -2x - 10 = -2(x + 5)
   $$
   This equation has all real solutions:
   $$
   x \in \mathbb{R}
   $$

4. **Fourth equation:** 
   $$
   -2(x - 5) = -2x - 10
   $$
   This equation has no solutions:
   $$
   x \in \varnothing
   $$

### Conclusion:
The first and second equations both yield $$ x = 0 $$, indicating that they are equivalent expressions. The third equation is valid for all real numbers, while the fourth has no solutions.

Thus, the pairs that show equivalent expressions are:
- $$ 2x + 10 = -2(x - 5) $$
- $$ -2(x + 5) = 2x - 10 $$
The first and second equations are equivalent, both giving $$ x = 0 $$.

Which pair shows equivalent expressions? \( 2 x+10=-2(x-5) \) \( -2(x+5)=2 x-10 \) \( -2 x-10=-2(x+5) \) \( -2(x-5)=-2 x-10 \)

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