https://file.aiquickdraw.com/mathbot/file/1726994569503r2ozxlhp.jpg

The equation in the image is:

\[
(x - 2)^2 - 5(x - 1)(x + 1) + (2x + 1)^2 = 10
\]

Let's solve this step-by-step.

### Step 1: Expand the terms
We need to expand each expression.

- For \((x - 2)^2\):
  \[
  (x - 2)^2 = x^2 - 4x + 4
  \]

- For \( -5(x - 1)(x + 1) \), use the difference of squares formula:
  \[
  (x - 1)(x + 1) = x^2 - 1
  \]
  So,
  \[
  -5(x^2 - 1) = -5x^2 + 5
  \]

- For \((2x + 1)^2\):
  \[
  (2x + 1)^2 = 4x^2 + 4x + 1
  \]

### Step 2: Combine all terms
Now substitute the expanded terms back into the equation:

\[
x^2 - 4x + 4 - 5(x^2 - 1) + 4x^2 + 4x + 1 = 10
\]

Simplify each part:

\[
x^2 - 4x + 4 - 5x^2 + 5 + 4x^2 + 4x + 1 = 10
\]

### Step 3: Combine like terms
Group the \(x^2\), \(x\), and constant terms:

- \(x^2 - 5x^2 + 4x^2 = 0\)
- \(-4x + 4x = 0\)
- \(4 + 5 + 1 = 10\)

So, the equation becomes:

\[
10 = 10
\]

### Conclusion:
Since the left-hand side equals the right-hand side, the equation is **an identity**, meaning it holds true for all values of \(x\). There is no specific solution because the equation is always true.

Would you like further clarification or more details on any step?

Here are some related questions:
1. What is an identity in algebra?
2. How do you solve quadratic equations?
3. Can an equation have no solutions?
4. How do you handle difference of squares?
5. What are some common forms of quadratic expansions?

**Tip:** Always simplify terms before solving equations to avoid errors!

(x - 2)^2 - 5(x - 1)(x + 1) + (2x + 1)^2 = 10

This page provides a detailed step-by-step solution to the quadratic equation (x - 2)^2 - 5(x - 1)(x + 1) + (2x + 1)^2 = 10, involving algebraic expansions and simplifications. Learn how to expand terms, apply the difference of squares, and simplify to find the solution.

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