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Prove that the equation
x^7+ 3x+ 3 = 0 has a unique solution.
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I assume that you are just Calculus.
Then the solution is simple.
Take the derivative of the left side function.
It is , and it is always positive (for all x).
It means that the left side function itself increases monotonically when x increases.
From the other side, the left side function is NEGATIVE when x tends to and POSITIVE when x tends to .
It implies that the root does exist and is unique.
To answer next question, simply look in your calculator (or computer software) for the plot of this polynomial.
When choose two consecutive integers in a way that the function is NEGATIVE at the smaller integer and is POSITIVE at the larger one.