for the function f, construct and simplify the difference quotient f(x + h)-f(x)/h f(x)=3x² + 4x

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=3(x + h)^2+4(x + h)=3(x^{2}+2xh+h^{2})+4x + 4h=3x^{2}+6xh+3h^{2}+4x + 4h$

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

$f(x + h)-f(x)=(3x^{2}+6xh+3h^{2}+4x + 4h)-(3x^{2}+4x)$ $=3x^{2}+6xh+3h^{2}+4x + 4h - 3x^{2}-4x=6xh+3h^{2}+4h$

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

$\frac{f(x + h)-f(x)}{h}=\frac{6xh+3h^{2}+4h}{h}=\frac{h(6x + 3h+4)}{h}=6x+3h + 4$

Answer:

$6x+3h + 4$