find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify.\n$f(x)=x^{2}+5x$\nquest…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=(x + h)^2+5(x + h)=x^{2}+2xh+h^{2}+5x + 5h$

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

$f(x + h)-f(x)=(x^{2}+2xh+h^{2}+5x + 5h)-(x^{2}+5x)$ $=x^{2}+2xh+h^{2}+5x + 5h - x^{2}-5x=2xh+h^{2}+5h$

Step3: Find the difference quotient

$\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}+5h}{h}$ Factor out $h$ from the numerator: $\frac{h(2x + h+5)}{h}$ Cancel out $h$ (assuming $h\neq0$): $2x+h + 5$

Answer:

$2x+h + 5$