Let $f(x)=x^3 - 3x + 1$ Let $f(x)=x^3 - 3x + 1$.   

a) Expand $(x+h)^3$.
Would this simply be $x^3 + h^3$?   

b. Use the formula $f’(x) =\lim_{h\to 0}\big(f(x+h)-f(x)\big)/h$ to show that the derivative of $f(x)$ is $3x^2-3$.
So, don’t we plug in $x^3 - 3x + 1$ every time we see $f(x)$ in that equation? 

c. find the intervals over which the graph of $f$ is increasing and decreasing. State the coordinates of any relative extreme values for the graph of $f$.   

for this, do we find the derivative of $f(x)=x^3 - 3x + 1$ which is $3x^2 - 3$ then solve for x and do a chart to find the positive and negative values. 

d. Find the intervals over which the graph of $f$ is concave up and concave down. State the coordinates of any points of inflection.  

For this, do we take the second derivative and do the same?
Based on the comments we need to begin at the beginning.

$(x+h)^3 = (x+h)(x+h)(x+h)$

Multiplication distributes over addition.

Lets start with one set of multiplications.

$(x+h)^2 = (x+h)(x+h)\\
(x+h)(x+h) = (x+h)x + (x+h)h\\
x\cdot x+h\cdot x  + x\cdot h+h\cdot h\\
(x+h)^2 = x^2 + 2xh + h^2$

Now lets work in the next factor.

$(x+h)^3 = (x+h)^2(x+h)\\
(x^2 + 2xh +h^2)(x+h)\\
(x^2 + 2xh +h^2)(x) +(x^2 + 2xh +h^2)(h)\\
x^3 + 2x^2h +xh^2+x^2h + 2xh^2 +h^3\\
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$

$f(x+h)$

We plug in $(x+h)$ everywhere we see $x$ in the definition of  $f(x)$

$f(x+h) = (x+h)^3 - 3(x+h) + 1$

I will let you multiply that out using the information above.

Next you will need to work out:

$f(x+h) - f(x) = (x+h)^3 - 3(x+h) + 1 - x^3 + 3x - 1$

c) what you describe is correct.  But without a) b) you can't find $f'(x)$

d) You are going to need, a second derivative!

Using the same processes you applied to $f(x)$ you will apply to $f'(x)$ to find

$\lim_\limits{h\to 0} \frac {f'(x+h) - f'(x)}{h}$
a) $(x+h)^3 = (x+h)(x+h)^2 = ? $

b) $f(x+h) = (x+h)^3-3(x+h)+1$. 

Use a) to expand and reduce. Then, compute $\frac{f(x+h)-f(x)}{h}$. When $h\to 0$, the expression you'll find should tend to $3x^2-3$.

c) and d) : yes and yes !