Find all integers $x$ such that $x^2+3x+24$ is a perfect square.
Now, do I find solution treating cases? But that doesn't seem very easy. Please help.
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Complete the square to get $(x+3/2)^2 + 87/4$. We want this to be a square itself, so $$(x+3/2)^2 + 87/4 = k^2.$$
This is the same as $$(2x+3)^2 + 87 = 4k^2.$$
Now consider the difference of squares $(2k - 2x - 3)(2k + 2x + 3) = 87 = 3\cdot 29$. Now there are only a few cases to check.
Do these give you any integer solutions for $x$?