Answer:
[tex]3x^2+(4+3h)x +(2h+h^2)[/tex]
Step-by-step explanation:
Evaluate the difference quotient in the usual way: put the function arguments where the variables are and simplify.
Difference quotient
The formula for the difference quotient is ...
[tex]\dfrac{f(x+h)-f(x)}{h}[/tex]
Application
For f(x) = x^3 +2x^2 -5, the difference quotient is ...
[tex]\dfrac{((x+h)^3+2(x+h)^2-5)-(x^3+2x^2-5)}{h}\\\\=\dfrac{x^3+3hx^2+3h^2x+h^3+2(x^2+2hx+h^2)-5-x^3-2x^2+5}{h}\\\\=\dfrac{(1-1)x^3+(3h+2-2)x^2+(3h^2+4h)x+(h^3+2h^2-5+5)}{h}\\\\=\dfrac{3hx^2+h(3h+4)x+h(h^2+2h)}{h}\\\\=\boxed{3x^2+(4+3h)x +(2h+h^2)}[/tex]
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Additional comment
We have shown h-terms with increasing powers to the right. That is because we're usually concerned with small values of h, so higher-degree terms become insignificant and can be neglected. If the expression were written in "standard form", it might be ...
h^2 +3hx +3x^2 +2h +4x . . . . . lexicographical order of decreasing degree
