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## To find the expression f(x,y+k)-f(x,y)/k, we need to substitute the given function f(x,y)=x^2y+2xy^2 into the expression and simplify it.

Let's start by substituting the function f(x,y) into the expression:

f(x, y+k) = (x^2)(y+k) + 2x(y+k)^2

= (x^2)(y+k) + 2x(y^2 + 2y*k + k^2)

= x^2y + x^2k + 2xy^2 + 4xyk + 2xk^2

Next, let's write out f(x, y) using the same function:

f(x, y) = x^2y + 2xy^2

Now, we can subtract f(x, y) from f(x, y+k):

f(x, y+k) - f(x, y) = (x^2y + x^2k + 2xy^2 + 4xyk + 2xk^2) - (x^2y + 2xy^2)

= x^2k + 4xyk + 2xk^2

Finally, divide the expression by k:

(f(x, y+k) - f(x, y))/k = (x^2k + 4xyk + 2xk^2)/k

= x^2 + 4xy + 2k

Therefore, the expression f(x,y+k)-f(x,y)/k simplifies to x^2 + 4xy + 2k.