The correct answer is option (D):
± 4
To determine the values of k that make the quadratic expression 9x2 + 3kx + 4 a perfect square, we need to understand the properties of a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In the general form, a perfect square trinomial can be written as (ax + b)2 = a2x2 + 2abx + b2 or (ax - b)2 = a2x2 - 2abx + b2.
Let's examine our given expression, 9x2 + 3kx + 4. We can see that the first term, 9x2, is a perfect square (3x)2 and the last term, 4, is also a perfect square (2)2. For the expression to be a perfect square trinomial, the middle term, 3kx, must be equal to ±2 times the product of the square roots of the first and last terms.
So we have:
* The square root of 9x2 is 3x.
* The square root of 4 is 2.
Therefore, the middle term should be ±2 * (3x) * 2 = ±12x.
Now, we can set up the equation:
3kx = ±12x
To find the value(s) of k, we can divide both sides by 3x:
k = ±12/3
k = ±4
So the values of k that make the expression a perfect square are +4 and -4. This gives us (3x + 2)2 = 9x2 + 12x + 4 or (3x - 2)2 = 9x2 - 12x + 4. Therefore the correct answer is ±4.