$$ \left(2x+1\right)^2-\left(x-1\right)^2 $$

To simplify the expression $ \left(2x + 1\right)^2 - \left(x - 1\right)^2 $, we can use the difference of squares formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, $ a = 2x + 1 $ and $ b = x - 1 $. Applying the difference of squares:

\[
\left(2x + 1\right)^2 - \left(x - 1\right)^2 = \left[(2x + 1) - (x - 1)\right]\left[(2x + 1) + (x - 1)\right]
\]

### Step 1: Simplify each binomial
- $ (2x + 1) - (x - 1) $ simplifies to:
  \[
  2x + 1 - x + 1 = x + 2
  \]
- $ (2x + 1) + (x - 1) $ simplifies to:
  \[
  2x + 1 + x - 1 = 3x
  \]

### Step 2: Multiply the simplified expressions
Now, multiply the two simplified binomials:

\[
(x + 2)(3x)
\]

Distribute $ 3x $ to both terms in $ x + 2 $:

\[
3x \cdot x + 3x \cdot 2 = 3x^2 + 6x
\]

### Final Answer:
\[
\left(2x + 1\right)^2 - \left(x - 1\right)^2 = 3x^2 + 6x
\]

Would you like to explore this further or ask about something else?

Here are 5 related questions:
1. How do you apply the difference of squares formula in other contexts?
2. What are the steps to expand and simplify $\left(x + y\right)^2$?
3. How would you solve $ \left(2x + 1\right)^2 = \left(x - 1\right)^2 $?
4. Can you explain the connection between binomial expansion and the difference of squares?
5. How would you graph the function $ f(x) = 3x^2 + 6x $?

**Tip:** Recognizing patterns like the difference of squares can simplify complex algebraic expressions quickly. Look for these patterns as a first step when simplifying.

Learn how to simplify the expression (2x + 1)^2 - (x - 1)^2 using the difference of squares formula. Step-by-step solution provided, including binomial expansion and algebraic simplification. Suitable for high school students studying algebra and polynomial expressions.

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