What prime is 4 greater than a perfect square and 7 less than the next-largest perfect square?

X - the prime
y² - the first perfect square
(y + 1)² - the second perfect square

(1) x = y² + 4
(2) x = (y + 1)² - 7

subtitute (2) to (1)

(y + 1)² - 7 = y² + 4    |use (a + b)² = a² + 2ab + b²

y² + 2y + 1 - 7 = y² + 4    |subtract y² from both sides

2y - 6 = 4    |add 6 to both sides

2y = 10    |divide both sides by 2

y = 5

subtitute the value of y to (1)

x = 5² + 4 = 25 + 4 = 29

check:
29 - it's prime /correct/
29 - 4 = 25 = 5² - the perfect square /correct/
29 + 7= 36 = 6² - the next-largest perfect square /correct/

29.