x+7)^9(x+8)(-4)^6<0

To solve the inequality \((x+7)^9(x+8)(-4)^6 < 0\), let's first analyze the components of the expression.

1. Identify the sign of each factor:

   • \((-4)^6\) is a positive constant because any even power of a negative number is positive.
   • The term \((x+7)^9\) is a polynomial that has a degree of 9, which means it will be:
     • Positive for \(x > -7\)
     • Zero at \(x = -7\)
     • Negative for \(x < -7\)
   • The term \((x+8)\) is a linear polynomial that will be:
     • Positive for \(x > -8\)
     • Zero at \(x = -8\)
     • Negative for \(x < -8\)

2. Find the critical points:

   • The critical points of the expression are where each factor is zero:
     • From \((x+7)^9\), we have \(x = -7\)
     • From \((x+8)\), we have \(x = -8\)

3. Test the intervals determined by the critical points:

   • The critical points divide the number line into the following intervals: \((-\infty, -8)\), \((-8, -7)\), and \((-7, \infty)\).

   • We will test a point in each interval to determine the sign of the expression in that interval.

   • Interval \( (-\infty, -8) \) (choose \(x = -9\)): \[ (-9+7)^9(-9+8) = (-2)^9(-1) = -(-2)^9 < 0 \] (Negative)

   • Interval \( (-8, -7) \) (choose \(x = -7.5\)): \[ (-7.5+7)^9(-7.5+8) = (-0.5)^9(0.5) = -0.5^{10} < 0 \] (Negative)

   • Interval \( (-7, \infty) \) (choose \(x = 0\)): \[ (0+7)^9(0+8) = 7^9(8) > 0 \] (Positive)

4. Conclusion:

   • The expression \((x+7)^9(x+8)(-4)^6 < 0\) holds true in the intervals where the product is negative, which is the interval \((-\infty, -8)\) and \((-8, -7)\).

Thus, the solution to the inequality is: \[ (-\infty, -8) \cup (-8, -7) \]