For the function f(x)=2x^2+x, calculate f(x+h) then calculate and simplify the difference quotient

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Well, we know that our function is f(x)=2x^2+x

Next, we calculate f(x+h). To do this, we simply put (x+h) into f(x)

f(x+h)=2(x+h)^2+(x+h)

f(x+h)= 2(x+h)(x+h)+(x+h)

f(x+h)=2(x^2+xh+xh+h^2)+(x+h)

= 2(x^2+2xh+h^2)+(x+h)

= 2x^2+4xh+2h^2+x+h

Now, we can use the difference quotient to find the slope

m = [ f (x + h) – f(x) ] / h

m= 2(x+h)^2 +(x+h)-2x^2+x

m= [(2x^2+4xh+2h^2+x+h)-(2x^2+x)]/ h

m= (2x^2+4xh+2h^2+x+h-2x^2-x)/ h

m= 4xh+2h^2+h / h

m= h(4x+2h+1)/ h

m= 4x+2h+1

Therefore the answer is

m=4x+2h+1

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