First, factor the numerator and denominator:
#(2x^2 + 11x + 5)/(3x^2 + 17x + 10) => ((2x + 1)(x + 5))/((3x + 2)(x + 5))#
Now, cancel common terms in the numerator and denominator:
#((2x + 1)color(red)(cancel(color(black)((x + 5)))))/((3x + 2)color(red)(cancel(color(black)((x + 5))))) => (2x + 1)/(3x + 2)#
To find the restrictions the denominator cannot be #0# therefore we need to solve for:
#3x^2 + 17x + 10 =#
Or
#(3x + 2)(x + 5) = 0#
Solution 1)
#3x + 2 = 0#
#3x + 2 - color(red)(2) = 0 - color(red)(2)#
#3x + 0 = -2#
#3x = -2#
#(3x)/color(red)(3) = -2/color(red)(3)#
#(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = -2/3#
#x = -2/3#
Solution 2)
#x + 5 = 0#
#x + 5 - color(red)(5) = 0 - color(red)(5)#
#x + 0 = -5#
#x = -5#
Therefore, the restrictions are:
#x != -2/3# and #x != -5#
