$f(x)$ is a quadratic polynomial (leading coefficient $>0$) and if $f(x)=0$ has two real roots $\alpha$ and $\beta$ such that $-1<\alpha<1<\beta<3$ and $f(0)

Because the leading coefficient is positive, the quadratic polynomial opens upwards, meaning it takes positive values outside the interval of the roots and negative values between them. We are given that alpha is between -1 and 1 while beta is between 1 and 3, so the roots lie on either side of x = 1 and x = 2. Since the condition f(0) < f(2) < 0 tells us the graph is still in the negative region at both x = 0 and x = 2, the root beta must be situated further to the right than 2. Consequently, x = 3 lies outside the roots where the graph is positive and rising, whereas x = -1 might still be negative or simply lower on the curve, ensuring that f(3) > f(-1).

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