Find the value of k for which x^2+4x+k it is perfect square.

if x^2 + d x

complete square
x^2 + d x + (d/2)^2
here
(4/2)^2 = 4
x^2 + 4 x + 4 = (x+2)^2

To find the value of k for which the quadratic expression x^2 + 4x + k is a perfect square, we need to determine the conditions that make it a perfect square.

A perfect square trinomial can be factored into a squared binomial. Let's assume the perfect square trinomial is (x + a)^2, where a is a constant. Expanding this expression, we get:
(x + a)^2 = x^2 + 2ax + a^2

Comparing this with the given trinomial x^2 + 4x + k, we can see that the coefficient of x is 2a, and the constant term is a^2. So, we have the following equations:

2a = 4 (coefficient of x is 4)
a^2 = k (constant term is k)

Solving the first equation for a, we find:
2a = 4
a = 2

Substituting the value of a into the second equation, we find:
a^2 = k
2^2 = k
4 = k

Therefore, the value of k for which x^2 + 4x + k is a perfect square is k = 4.