Let's first analyze the problem from the image.

From the provided information:

• The point M is between L and N on line LN.
• The distances are provided as:
  • LM = $x^2$
  • MN = $x$
  • LN = 12 (the entire length from L to N).

The relationship we have here is the sum of LM and MN equals LN: $LM + MN = LN$

Substitute the given values into the equation: $x^2 + x = 12$

Now we need to solve the quadratic equation: $x^2 + x - 12 = 0$

Solving the quadratic equation:

We will solve $x^2 + x - 12 = 0$ using factoring, the quadratic formula, or completing the square. Let's start with factoring.

We need to find two numbers that multiply to $-12$ and add up to $1$. These numbers are $4$ and $-3$.

So, the equation factors as: $(x + 4)(x - 3) = 0$

Setting each factor equal to zero: $x + 4 = 0 \quad \text{or} \quad x - 3 = 0$

Thus, the solutions for $x$ are: $x = -4 \quad \text{or} \quad x = 3$

Checking the solution:

Since distances cannot be negative, we discard $x = -4$ and keep $x = 3$.

Final Answer:

• $LM = x^2 = 3^2 = 9$
• $MN = x = 3$

Let me know if you need further explanation or have any other questions! Here are some follow-up questions to help reinforce your understanding:

1. What is the method for solving a quadratic equation if factoring doesn't work?
2. How would the result change if $x = -4$ was valid in the context of this problem?
3. Can the distances (LM and MN) be any value, or are they restricted by the geometry of the problem?
4. What does it mean for a point to be "between" two others on a line segment in terms of geometry?
5. How can we apply this method to solve similar problems in geometry?

Tip: When solving a quadratic equation, always check for possible extraneous solutions that might not fit the physical context of the problem!