First, rewrite this expression as:
#(18x^4 - 14x^3 + 8x^2 + 7)/(2x^2)#
We can then rewrite this again as:
#(18x^4)/(2x^2) - (14x^3)/(2x^2) + (8x^2)/(2x^2) + 7/(2x^2)#
We can once again rewrite this as:
#(9x^4)/x^2 - (7x^3)/x^2 + (4x^2)/x^2 + 7/(2x^2)#
We can now use these rule for exponents to divide the #x# terms in the first three fractions:
#x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))# and #a^color(red)(1) = a# and #a^color(red)(0) = 1#
#(9x^color(red)(4))/x^color(blue)(2) - (7x^color(red)(3))/x^color(blue)(2) + (4x^color(red)(2))/x^color(blue)(2) + 7/(2x^2) =>#
#9x^(color(red)(4)-color(blue)(2)) - 7x^(color(red)(3)-color(blue)(2)) + 4x^(color(red)(2)-color(blue)(2)) + 7/(2x^2) =>#
#9x^2 - 7x^1 + 4x^0 + 7/(2x^2) =>#
#9x^2 - 7x + 4 + 7/(2x^2)#
Or, we can use this rule of exponents to eliminate the exponent in the denominator of the last term:
#1/x^color(red)(a) = x^color(red)(-a)#
#9x^2 - 7x + 4 + 7/(2x^color(red)(2)) =>#
#9x^2 - 7x + 4 + 7/2x^color(red)(-2)#
