Answer:
[tex]-4x-2h-3[/tex]
Explanations:
Given the function
[tex]f(x)=-2x^2-3x+6[/tex]
We are to look for the difference quotient:
[tex]\frac{f(x+h)-f(x)}{h}[/tex]
Get f(x + h)
[tex]\begin{gathered} f(x+h)=-2(x+h)^2-3(x+h)+6 \\ f(x+h)=-2(x^2+2xh+h^2)-3x-3h+6 \\ f(x+h)=-2x^2-4xh-2h^2-3x-3h+6 \\ \end{gathered}[/tex]
Given f(x) expressed as:
[tex]f(x)=-2x^2-3x+6[/tex]
Substitute both functions into the difference quotient;
[tex]\begin{gathered} \frac{-2x^2-4xh-2h^2-3x-3h+6-(-2x^2-3x+6)}{h} \\ \frac{-\cancel{2x^2}-4xh-2h^2-\cancel{3x}-3h+\cancel{6}+\cancel{2x^2}+\cancel{3x}-\cancel{6}}{h} \\ \frac{-4xh-2h^2-3h}{h} \end{gathered}[/tex]
Factor out "h" from the result to have
[tex]\begin{gathered} \frac{\cancel{h}(-4x-2h-3)}{\cancel{h}} \\ -4x-2h-3 \end{gathered}[/tex]
This shows that the simplified form of the expression is -4x - 2h - 3
