## The Misconception: Why (k+4)^2 ≠ k^2 + 4^2

It's a common mistake to assume that squaring a sum is the same as squaring each term individually. However, **(k+4)^2 is not equal to k^2 + 4^2**. Let's explore why this is the case and how to correctly expand the expression.

### Understanding the Difference

**The expression (k+4)^2 represents squaring the entire binomial (k+4)**. This means multiplying the entire expression by itself:

**(k+4)^2 = (k+4)(k+4)**

To correctly expand this expression, we need to apply the distributive property:

**Multiply k by each term inside the second set of parentheses:**k*k + k*4**Multiply 4 by each term inside the second set of parentheses:**4*k + 4*4

This gives us:

**k^2 + 4k + 4k + 16**

Combining like terms, we get the final result:

**(k+4)^2 = k^2 + 8k + 16**

### Why the Misconception Occurs

The misconception likely arises from the assumption that squaring a sum follows the same rules as multiplication with a single variable:

**For example, 2*(a+b) = 2a + 2b

However, squaring involves multiplying a quantity by itself, which introduces additional terms.

### The Importance of Expanding Correctly

Understanding how to correctly expand expressions like (k+4)^2 is crucial for solving algebraic equations, simplifying expressions, and working with quadratic functions. Using the incorrect formula can lead to inaccurate solutions and flawed conclusions.

Remember, **always apply the distributive property and combine like terms when expanding expressions involving squared binomials.**