To solve for the difference quotient of the given function [tex]\( f(x) = -9x^2 + 6x + 6 \)[/tex] and subsequently simplify it, follow these steps:
1. Define the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -9x^2 + 6x + 6 \][/tex]
2. Determine [tex]\( f(x + h) \)[/tex] by substituting [tex]\( x + h \)[/tex] into the function:
[tex]\[ f(x + h) = -9(x + h)^2 + 6(x + h) + 6 \][/tex]
[tex]\[ = -9(x^2 + 2xh + h^2) + 6x + 6h + 6 \][/tex]
[tex]\[ = -9x^2 - 18xh - 9h^2 + 6x + 6h + 6 \][/tex]
3. Compute the difference [tex]\( f(x + h) - f(x) \)[/tex]:
[tex]\[ f(x + h) - f(x) = (-9x^2 - 18xh - 9h^2 + 6x + 6h + 6) - (-9x^2 + 6x + 6) \][/tex]
[tex]\[ = -9x^2 - 18xh - 9h^2 + 6x + 6h + 6 + 9x^2 - 6x - 6 \][/tex]
[tex]\[ = -18xh - 9h^2 + 6h \][/tex]
4. Form the difference quotient by dividing by [tex]\( h \)[/tex]:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{-18xh - 9h^2 + 6h}{h} \][/tex]
[tex]\[ = \frac{-18xh}{h} + \frac{-9h^2}{h} + \frac{6h}{h} \][/tex]
[tex]\[ = -18x - 9h + 6 \][/tex]
5. Simplify the expression (since [tex]\( h \neq 0 \)[/tex]):
[tex]\[ \frac{f(x + h) - f(x)}{h} = -18x - 9h + 6 \][/tex]
Therefore, the simplified difference quotient for the function [tex]\( f(x) = -9x^2 + 6x + 6 \)[/tex] is:
[tex]\[ \boxed{-18x - 9h + 6} \][/tex]
