Find a factored form of the given algebraic expression involving squared terms and cross terms | Step-by-Step Solution

1. What We're Solving:

We need to factorize the expression: -a²-b²-c²+2ab+2bc+2ac

This means we want to rewrite it as a product of simpler expressions (like finding what multiplies together to give us this result).

2. The Approach:

This expression has a special pattern - it looks like it might be related to the square of a sum or difference. When we see squared terms mixed with cross-product terms (like 2ab, 2bc, 2ac), it often hints at a perfect square trinomial pattern, but extended to three variables.

The key insight is to recognize that this expression is the negative of the expansion of (a+b+c)². Let's explore this step by step!

3. Step-by-Step Solution:

Step 1: Let's first recall what happens when we expand (a+b+c)² (a+b+c)² = (a+b+c)(a+b+c) = a² + b² + c² + 2ab + 2bc + 2ac

Step 2: Now look at our original expression: -a²-b²-c²+2ab+2bc+2ac

Let's rearrange it to see the pattern more clearly: = -(a² + b² + c²) + (2ab + 2bc + 2ac) = -(a² + b² + c² - 2ab - 2bc - 2ac)

Step 3: Notice what's inside the parentheses now: a² + b² + c² - 2ab - 2bc - 2ac

Step 4: The correct factorization is: -a²-b²-c²+2ab+2bc+2ac = -(a-b-c)(a+b-c)

4. The Answer:

The factored form is: -(a-b-c)(a+b-c) or equivalently (b+c-a)(a+b-c)

(You can verify this by expanding either form back to the original expression!)

5. Memory Tip:

When you see an expression with all squared terms and all cross-product terms, think "perfect square pattern!" The negative sign in front suggests we're looking at the negative of a perfect square or a difference of squares pattern. These symmetric expressions often factor beautifully once you spot the pattern!

Great job working through this challenging factorization - these multi-variable expressions can be tricky, but recognizing patterns is the key! 🌟