Question 905095
{{{f(x) = -3x^2 + 3x + 1}}}

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First we need to find f(x+h)


{{{f(x) = -3x^2 + 3x + 1}}}


{{{f(x+h) = -3(x+h)^2 + 3(x+h) + 1}}} Replace every x with x+h. Then expand.


{{{f(x+h) = -3(x^2+2xh+h^2) + 3(x+h) + 1}}}


{{{f(x+h) = -3x^2-6xh-3h^2 + 3x+3h + 1}}}


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Now we plug in {{{f(x+h) = -3x^2-6xh-3h^2 + 3x+3h + 1}}} and {{{f(x) = -3x^2 + 3x + 1}}} into {{{(f(x+h)-f(h))/h}}}


Simplify as much as possible


{{{(f(x+h)-f(h))/h = ((-3x^2-6xh-3h^2 + 3x+3h + 1) - (-3x^2 + 3x + 1))/h}}}


{{{(f(x+h)-f(h))/h = (-3x^2-6xh-3h^2 + 3x+3h + 1 +3x^2 - 3x - 1)/h}}}


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


{{{(f(x+h)-f(h))/h = (h(-6x-3h+3))/h}}}


{{{(f(x+h)-f(h))/h = (highlight(h)(-6x-3h+3))/(highlight(h))}}}


{{{(f(x+h)-f(h))/h = (cross(h)(-6x-3h+3))/(cross(h))}}}


{{{(f(x+h)-f(h))/h = -6x-3h+3}}}


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Finally, we plug in x = 4


{{{(f(x+h)-f(h))/h = -6x-3h+3}}}


{{{(f(4+h)-f(h))/h = -6(4)-3h+3}}}


{{{(f(4+h)-f(h))/h = -24-3h+3}}}


{{{(f(4+h)-f(h))/h = -21-3h}}}


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The final answer is {{{(f(4+h)-f(h))/h = -21-3h}}}