To solve this, we calculate the areas of the triangle and the rectangle separately and find their difference.

Step 1: Area of the triangle

The formula for the area of a triangle is:

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}.$

Substituting the given values: $\text{Area of triangle} = \frac{1}{2} \times (3y - 1) \times 6y.$

Simplify: $\text{Area of triangle} = \frac{1}{2} \times (18y^2 - 6y) = 9y^2 - 3y.$

Step 2: Area of the rectangle

The formula for the area of a rectangle is:

$\text{Area of rectangle} = \text{base} \times \text{height}.$

Substituting the given values: $\text{Area of rectangle} = (y + 4) \times (5y - 7).$

Simplify: $\text{Area of rectangle} = y(5y - 7) + 4(5y - 7),$ $\text{Area of rectangle} = 5y^2 - 7y + 20y - 28,$ $\text{Area of rectangle} = 5y^2 + 13y - 28.$

Step 3: Difference between the areas

Now subtract the area of the rectangle from the area of the triangle:

$\text{Difference} = (\text{Area of triangle}) - (\text{Area of rectangle}),$ $\text{Difference} = (9y^2 - 3y) - (5y^2 + 13y - 28).$

Simplify: $\text{Difference} = 9y^2 - 3y - 5y^2 - 13y + 28,$ $\text{Difference} = (9y^2 - 5y^2) + (-3y - 13y) + 28,$ $\text{Difference} = 4y^2 - 16y + 28.$

Final Answer:

The expression representing the difference in terms of $y$ is: $\boxed{4y^2 - 16y + 28.}$

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Would you like further explanation of any step? Here are related questions you may find helpful:

1. How do you find the area of a triangle or a rectangle?
2. What is the general method for expanding and simplifying polynomial expressions?
3. How do you determine whether the resulting difference is positive or negative for given $y$ values?
4. Can this difference be factored further?
5. What are the practical applications of comparing areas in terms of algebraic expressions?

Tip: Always double-check your signs when subtracting polynomials, as errors can easily occur.