Let's solve this step by step.

Step 1: Define the variables

Let the two consecutive negative even integers be $x$ and $x + 2$.

Step 2: Express the sum of the squares of the integers

The sum of their squares is given by: $x^2 + (x + 2)^2 = 100$

Step 3: Expand the equation

Expand $(x + 2)^2$: $x^2 + (x^2 + 4x + 4) = 100$ Simplify the equation: $2x^2 + 4x + 4 = 100$

Step 4: Move everything to one side

Subtract 100 from both sides: $2x^2 + 4x + 4 - 100 = 0$ Simplify: $2x^2 + 4x - 96 = 0$

Step 5: Simplify the equation

Divide the entire equation by 2: $x^2 + 2x - 48 = 0$

Step 6: Solve the quadratic equation

We can now solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $x^2 + 2x - 48 = 0$, $a = 1$, $b = 2$, and $c = -48$. Substituting these values into the quadratic formula: $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-48)}}{2(1)}$ $x = \frac{-2 \pm \sqrt{4 + 192}}{2}$ $x = \frac{-2 \pm \sqrt{196}}{2}$ $x = \frac{-2 \pm 14}{2}$

Step 7: Find the two possible values for $x$

Now, solve for $x$:

1. $x = \frac{-2 + 14}{2} = \frac{12}{2} = 6$
2. $x = \frac{-2 - 14}{2} = \frac{-16}{2} = -8$

Step 8: Interpret the results

Since the integers are negative, we discard $x = 6$ and keep $x = -8$.

So, the two consecutive negative even integers are $-8$ and $-6$.

Step 9: Verify the solution

Check the sum of the squares: $(-8)^2 + (-6)^2 = 64 + 36 = 100$ The sum of the squares is indeed 100.

Final Answer:

The two consecutive negative even integers are $\boxed{-8}$ and $\boxed{-6}$.

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Would you like any further details on the solution or have any questions about this process?

Here are 5 related questions:

1. How would the solution change if the sum of the squares were a different number, say 144?
2. How do we solve a quadratic equation if the discriminant is negative?
3. What are other methods for solving quadratic equations besides the quadratic formula?
4. How can we find two consecutive odd integers that satisfy a similar sum of squares condition?
5. What are the general steps for solving word problems involving consecutive integers?

Tip: For quadratic equations, always check if the discriminant (the expression under the square root) is positive, zero, or negative, as this affects the nature of the roots!