for the function f(x) = 5x² + 3x + 1, evaluate and fully simplify each of the following. f(x + h) = f(x + h)…

Explanation:

Step1: Substitute $x + h$ into $f(x)$

[ \begin{align*} f(x + h)&=5(x + h)^2+3(x + h)+1\ &=5(x^{2}+2xh+h^{2})+3x + 3h+1\ &=5x^{2}+10xh+5h^{2}+3x + 3h+1 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}+3x + 3h+1)-(5x^{2}+3x + 1)\ &=5x^{2}+10xh+5h^{2}+3x + 3h+1 - 5x^{2}-3x - 1\ &=10xh+5h^{2}+3h \end{align*} ]

Step3: Calculate $\frac{f(x + h)-f(x)}{h}$

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}+3h}{h}\ &=\frac{h(10x + 5h+3)}{h}\ &=10x + 5h+3 \end{align*} ]

Answer:

$f(x + h)=5x^{2}+10xh+5h^{2}+3x + 3h+1$ $\frac{f(x + h)-f(x)}{h}=10x + 5h+3$