Reduce each rational expression to lowest terms:

3x^2-12y^2 over x^2+4xy+4y^2

(3x^2 - 12y^2) / (x^2 + 4xy + 4y^2),

3(x^2 - 4y^2) / (x + 2y) (x + 2y),
3(x + 2y) (x - 2y) / (x + 2y)(x - 2y),
Cancel two (x + 2y) factors and get:

3(x - 2y) / (x + 2y).

CORRECTION: The denominator of the 3rd expression should be (x + 2y) (x + 2y).

To reduce the rational expression (3x^2-12y^2) / (x^2+4xy+4y^2) to the lowest terms, we need to factor both the numerator and denominator and cancel out any common factors.

First, let's factor the numerator and denominator.

The numerator is a difference of squares, so we can apply the formula (a^2 - b^2) = (a + b)(a - b):
3x^2 - 12y^2 = 3(x^2 - 4y^2) = 3(x + 2y)(x - 2y)

The denominator is a perfect square trinomial, so we can rewrite it as a square of a binomial:
x^2 + 4xy + 4y^2 = (x + 2y)^2

Now, we can cancel out the common factor of (x + 2y) in the numerator and denominator:
(3(x + 2y)(x - 2y)) / ((x + 2y)^2)

After canceling out the common factor, the reduced expression is:
3(x - 2y) / (x + 2y)

To reduce a rational expression to its lowest terms, you need to simplify both the numerator and denominator as much as possible. In this case, we have the rational expression:

(3x^2 - 12y^2) / (x^2 + 4xy + 4y^2)

To simplify this expression, start by factoring the numerator and denominator.

First, factor out the greatest common factor in the numerator, which is 3:

3(x^2 - 4y^2)

The numerator can be further factored using the difference of squares formula: a^2 - b^2 = (a + b)(a - b):

3(x + 2y)(x - 2y)

Now let's factor the denominator:

x^2 + 4xy + 4y^2

This expression is a perfect square trinomial and can be factored as:

(x + 2y)^2

Now, we can rewrite the rational expression in factored form:

3(x + 2y)(x - 2y) / (x + 2y)^2

Notice that the factor (x + 2y) appears in both the numerator and the denominator. You can cancel out this common factor:

3(x - 2y) / (x + 2y)

This is the reduced form of the rational expression.