Look at #(x + 1)(x - 3) = 0

This means that the function is 0 at #x = -1 and x = 3#.

Also, it means that the sign of the corresponding factor changes sign at that value of x.

At values of x < -1:

Both #(x + 1)# and #(x - 3)# are negative. A negative multiplied by a negative is a positive, therefore, x < -1 is one of the regions where #(x + 1)(x - 3) > 0#. Let's make a note of that:

#x < -1#

At values between -1 and 3:

#(x + 1)# is positive but #(x -3)# is still negative. A positive multiplied by a negative is negative, therefore, this is NOT a region for #(x + 1)(x - 3) >0#

At values #x > 3#:

Both #(x + 1)# and #(x - 3)# are positive. A positive multiplied by a positive is a positive, therefore, x > 3 is one of the regions where #(x + 1)(x - 3) > 0. Let's make a note of that:

#x < -1 and x > 3#

We have no more regions to investigate, therefore, the above is our answer.