find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify it. $f(x)=x^{2}+7x + 4$…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=(x + h)^2+7(x + h)+4=x^{2}+2xh+h^{2}+7x + 7h+4$

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}+7x + 7h+4)-(x^{2}+7x + 4)\ &=x^{2}+2xh+h^{2}+7x + 7h+4 - x^{2}-7x - 4\ &=2xh+h^{2}+7h \end{align*} ]

Step3: Find the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}+7h}{h}\ &=\frac{h(2x + h+7)}{h}\ &=2x+h + 7 \end{align*} ]

Answer:

$2x+h + 7$