The product of two consecutive integers is represented as $x(x + 1)$, where $x$ is the lesser number.

Given that the product of these integers equals 420, the equation becomes:

$x(x + 1) = 420$

Expanding the left-hand side:

$x^2 + x = 420$

This corresponds to the quadratic equation $x^2 + x = 420$.

Thus, the correct equation is:

$\boxed{x^2 + x = 420}$

Would you like further clarification on this?

Here are 5 related questions:

1. How do you solve the quadratic equation $x^2 + x = 420$?
2. What are consecutive integers, and how do they relate to quadratic equations?
3. Can a quadratic equation have more than two solutions?
4. How do you factor quadratic equations like $x^2 + x - 420 = 0$?
5. What are some other real-world examples of quadratic equations?

Tip: When dealing with consecutive numbers, always try to express them as $x$ and $x+1$, or $x$ and $x-1$.