$300x^{299}+299x^4+343x^3+23x+300=0$

If $\alpha$ is the real root of the equation above, find $\lfloor \alpha \rfloor$ .

The answer is -1.

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We can do this by eradicating cases.

If $\alpha>0$ we see that whole expression always remains positive.

If $\alpha<-1$ we see that $300x^{299}$ becomes too much large for the expression to be zero , making the polynomial always negative.

So we can easily conclude that $-1<\alpha<0$ and hence $\lfloor \alpha \rfloor = -1$