Question

follow the step - by - step process in parts a through c to determine the difference quotient, $f(x)=2x^{2}+5x - 3$. a. determine $f(x + h)$. $f(x + h)=square$ (simplify your answer. do not factor.) b. simplify the expression $f(x + h)-f(x)$. $f(x + h)-f(x)=square$ (simplify your answer. do not factor.) c. determine the difference quotient $\frac{f(x + h)-f(x)}{h}$.

Explanation:

Step1: Find f(x + h)

Substitute \(x+h\) into \(f(x)\):
\[

$$\begin{align*} f(x + h)&=2(x + h)^2+5(x + h)-3\\ &=2(x^{2}+2xh+h^{2})+5x + 5h-3\\ &=2x^{2}+4xh+2h^{2}+5x + 5h-3 \end{align*}$$

\]

Step2: Find f(x + h)-f(x)

\[

$$\begin{align*} f(x + h)-f(x)&=(2x^{2}+4xh+2h^{2}+5x + 5h-3)-(2x^{2}+5x - 3)\\ &=2x^{2}+4xh+2h^{2}+5x + 5h-3 - 2x^{2}-5x + 3\\ &=4xh+2h^{2}+5h \end{align*}$$

\]

Step3: Find the difference quotient \(\frac{f(x + h)-f(x)}{h}\)

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{4xh+2h^{2}+5h}{h}\\ &=\frac{h(4x + 2h+5)}{h}\\ &=4x+2h + 5 \end{align*}$$

\]

Answer:

a. \(2x^{2}+4xh+2h^{2}+5x + 5h-3\)
b. \(4xh+2h^{2}+5h\)
c. \(4x+2h + 5\)