Question 356853
<pre>

Some books use "<font face = "symbol">D</font>x" and some use "h" to find the difference quotient.
I will use "h".  If your book uses "<font face = "symbol">D</font>x" then 

{{{f(x) = -6x^2+7x+4}}}

Difference quotient = {{{(f(x+h)-f(x))/h}}}

First find the term on the upper left {{{f(x+h)}}} and simplify:

{{{f(x+h) = -6(x+h)^2+7(x+h)+4}}}
{{{f(x+h) = -6(x+h)(x+h)+7x+7h+4}}}
{{{f(x+h) = -6(x^2+2hx+h^2)+7x+7h+4}}}
{{{f(x+h) = -6x^2-12hx-6h^2+7x+7h+4}}}

Substitute {{{(-6x^2-12hx-6h^2+7x+7h+4)}}} for {{{f(x+h)}}} and {{{(-6x^2+7x+4)}}} for {{{f(x)}}} in
 
{{{(f(x+h)-f(x))/h}}}

{{{((-6x^2-12hx-6h^2+7x+7h+4)-(-6x^2+7x+4))/h}}}

Remove parentheses:

{{{(-6x^2-12hx-6h^2+7x+7h+4+6x^2-7x-4)/h}}}

Cancel what will cancel:

{{{(cross(-6x^2)-12hx-6h^2+cross(7x)+7h+cross(4)+cross(6x^2)-cross(7x)-cross(4))/h}}}

{{{(-12hx-6h^2+7h)/h}}}

Factor h out of the top:

{{{(h(-12x-6h+7))/h}}}

Cancel the h's

{{{(cross(h)(-12x-6h+7))/cross(h)}}}

{{{-12x-6h+7}}}

Edwin</pre>