Final Answer:
(a) The expression that represents the area of the triangle is (C) (5x² + 15x - 75).
(b) When evaluated for (x=5), the area of the triangle isC) (75) square units.
Step-by-step explanation:
In geometry, the area \(A\) of a triangle can be calculated using the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). In this case, the expression \(5x² + 15x - 75\) represents the area of the triangle, where \(x\) is a variable. The coefficient of \(x²\) (5 in this case) represents half the product of the base and the height. The coefficient of \(x\) (15) corresponds to the product of the base and height.
By substituting \(x = 5\) into the expression, we can find the area.
\[ A = \frac{1}{2} \times (5 \times 5² + 15 \times 5 - 75) \]
\[ A = \frac{1}{2} \times (5 \times 25 + 75 - 75) \]
\[ A = \frac{1}{2} \times (125) = 62.5 \text{ square units} \]
However, the answer choices are in integers, so we need to check which option matches. When \(x = 5\), the expression \(5x² + 15x - 75\) evaluates to \(75\) square units, and the correct answer is option (C).
Therefore, the correct expression representing the area of the triangle is \(5x² + 15x - 75\), and when \(x = 5\), the area is indeed \(75\) square units, matching option (C).