solve the inequality express in interval notation

Question

x^4 + 4x^3 - 9x^2 - 36x < 0

Answer 1

x(x^3 + 4x^2 - 9x - 36) < 0
x( x^2(x+4) - 9(x+4) ) = 0
x(x+4)(x^2-9) < 0
x(x+4)(x+3)(x-3) < 0

so your critical values are
x = 0, -3, -4, and 3

Knowing that y = x^4 + .... is a quartic that rises in the 1st and 2nd quadrant because of the +1x^4 term, and it has 4 x-intercepts, you can make a rough sketch showing a W shape with those intercepts.

It should be obvious that the graph is below the x-axis ( y < 0 ) for values between -3 and -4 or between 0 and 3
so in standard notation:
-4 < x < -3 OR 0 < x < 3

I will leave it up to you to express it in interval notation

Answer 2

Thank you!