given f(x)=x² + 6x, (a) find f(x + h) and simplify. (b) find (f(x + h)-f(x))/h and simplify. part: 0 / 5…

Explanation:

Step1: Expand $(x + h)^2$

Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a = x$ and $b = h$, we have $(x + h)^2=x^{2}+2xh+h^{2}$. So $f(x + h)=(x + h)^2+6(x + h)=x^{2}+2xh+h^{2}+6x + 6h$.

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

$f(x)=x^{2}+6x$, so $f(x + h)-f(x)=(x^{2}+2xh+h^{2}+6x + 6h)-(x^{2}+6x)=x^{2}+2xh+h^{2}+6x + 6h - x^{2}-6x=2xh+h^{2}+6h$.

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

$\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}+6h}{h}$. Factor out $h$ from the numerator: $\frac{h(2x + h+6)}{h}$. Cancel out the $h$ terms, we get $2x+h + 6$.

Answer:

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