The problem asks to find Carol's age based on the condition:

"If you double the square of my age and add my age to it, the sum is less than or equal to 210."

Let's define Carol's age as $x$. According to the given condition, the equation is:

$2x^2 + x \leq 210$

To find the possible values of Carol's age, we will solve this inequality step by step.

Steps:

1. Start with the inequality: $2x^2 + x \leq 210$

2. Rearrange it: $2x^2 + x - 210 \leq 0$

3. Solve the quadratic inequality by first solving the equation $2x^2 + x - 210 = 0$ using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 2$, $b = 1$, and $c = -210$.

Let's calculate the roots next.The quadratic equation $2x^2 + x - 210 = 0$ has two solutions:

$x_1 = 10 \quad \text{and} \quad x_2 = -10.5$

Since age cannot be negative, Carol's age must be:

$x = 10$

Thus, Carol could be 10 years old.

Would you like more details or further explanations? Here are some related questions to deepen your understanding:

1. How do we solve quadratic inequalities like the one in this problem?
2. What does the quadratic formula represent geometrically?
3. How would the solution change if the inequality was strictly $2x^2 + x < 210$?
4. Can we verify that $x = 10$ satisfies the original condition?
5. Why do we discard the negative root in this context?

Tip: When solving inequalities, always remember to check if the solution fits the practical context, such as ignoring negative values for age!