You made a small mistake in your calculation. When completing the square, you correctly added 9/2 to the expression, but you forgot to subtract 9/2 after multiplying by 4. Here's the corrected working:
Factor out the coefficient of x^2, which is 4:
4(x^2 + 3x) + 7
Take half of the coefficient of x, which is 3, and square it to get 9/4:
4(x^2 + 3x + 9/4) + 7 - 9
Rewrite the first three terms as a perfect square:
4(x + 3/2)^2 - 2
So the correct answer is 4(x + 3/2)^2 - 2, which is the same as the expression you got, but without the extra factor of 4/4.