Number of Negative Integer Solutions

The function $f(x)$ has 48 roots: $-4, -6, -8, \dots, -98$. For $x > -4$ or $x < -98$, all 48 factors have the same sign (all positive or all negative), so $f(x) > 0$. $f(x) < 0$ when an odd number of factors are negative. This occurs in intervals between odd roots:

• $(-6, -4)$: $x = -5$ (1 value)
• $(-10, -8)$: $x = -9$ (1 value)
• ...
• $(-98, -96)$: $x = -97$ (1 value) These form an AP: $-5, -9, -13, \dots, -97$. Number of terms $= \frac{-97 - (-5)}{-4} + 1 = 24$.