To determine the equation of a circle with a given center and radius, we use the standard form of the equation of a circle.
The general form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given that the center of the circle is [tex]\((-2, 3)\)[/tex] and the radius is [tex]\(4\)[/tex], we substitute these values into the equation.
The values are:
[tex]\[
h = -2, \quad k = 3, \quad r = 4
\][/tex]
Substituting [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the general form:
[tex]\[
(x - (-2))^2 + (y - 3)^2 = 4^2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
(x + 2)^2 + (y - 3)^2 = 4^2
\][/tex]
We know that [tex]\(4^2 = 16\)[/tex], so the equation becomes:
[tex]\[
(x + 2)^2 + (y - 3)^2 = 16
\][/tex]
Thus, the correct equation representing the circle is:
[tex]\[
(x + 2)^2 + (y - 3)^2 = 16
\][/tex]
Among the given options, the equation that matches this form is:
[tex]\[
\text{D. } (x + 2)^2 + (y - 3)^2 = 16
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
