## Understanding the Proof of (a + b)^2 = a^2 + b^2 + 2ab

The equation **(a + b)^2 = a^2 + b^2 + 2ab** is a fundamental algebraic identity that holds true for all values of *a* and *b*. This identity is often used to simplify expressions, solve equations, and understand the relationship between squares and products.

Here's a breakdown of the proof:

### Expanding the Left-Hand Side

The left-hand side of the equation is **(a + b)^2**. This means we are squaring the binomial *(a + b)*.

**Recall that squaring a term means multiplying it by itself:**

(a + b)^2 = (a + b) * (a + b)

### Applying the Distributive Property

Now, we use the distributive property to expand the product. The distributive property states that the product of a sum and a number is equal to the sum of the products of the number and each term in the sum.

(a + b) * (a + b) = a * (a + b) + b * (a + b)

### Further Expansion

We distribute again to expand the right-hand side:

a * (a + b) + b * (a + b) = a * a + a * b + b * a + b * b

### Simplifying

Finally, we simplify the expression by combining like terms and using the commutative property of multiplication (a * b = b * a):

a * a + a * b + b * a + b * b = **a^2 + 2ab + b^2**

### Conclusion

We have shown that **(a + b)^2** can be expanded and simplified to **a^2 + b^2 + 2ab**. This proves the identity, demonstrating that the two sides of the equation are equivalent for any values of *a* and *b*.

This proof highlights the power of algebraic manipulation and the importance of understanding fundamental identities in mathematics.