Answer:
[tex] 2x + 5 + h [/tex]
Step-by-step explanation:
You are told f(x) is defined this way.
[tex] f(x) = x^2 + 5x - 3 [/tex]
To find f(x + h), replace x with x + h in the definition of the function.
[tex] f(x + h) = (x + h)^2 + 5(x + h) - 3 [/tex] Equation 1
Now you need to insert this entire expression in the fraction you are given and simplify it.
Here is the expression you were given:
[tex] \dfrac{f(x + h) - f(x)}{h} [/tex]
Now we replace f(x + h) with the expression above (Eq. 1). We replace f(x) with the original quadratic expression.
[tex] \dfrac{(x + h)^2 + 5(x + h) - 3 - (x^2 + 5x - 3)}{h} [/tex]
Now we expand the square of the binomial and distribute the negative throughout the parentheses.
[tex] \dfrac{x^2 + 2hx + h^2 + 5x + 5h - 3 - x^2 - 5x + 3)}{h} [/tex]
2x + h + 5
Combine like terms in the numerator.
[tex] \dfrac{2hx + h^2 + 5h)}{h} [/tex]
[tex] \dfrac{h(2x + h + 5)}{h} [/tex]
[tex] 2x + 5 + h [/tex]
