Calculate Trapezoid Area: Solving with Height h=(12x-8) and Parallel Sides

To solve this problem, we apply the formula for the area of a trapezoid:

• Step 1: Identify and substitute the expressions for the bases and height:
  $b_1 = x + 5$, $b_2 = 13x - 2$, $h = 12x - 8$.
• Step 2: Calculate $b_1 + b_2$:
  $b_1 + b_2 = (x + 5) + (13x - 2) = 14x + 3$.
• Step 3: Substitute into the area formula:
  $A = \frac{1}{2} \times (14x + 3) \times (12x - 8)$.
• Step 4: Distribute and simplify:
  $A = \frac{1}{2} \times ((14x + 3) \cdot (12x - 8))$.
• Step 5: Expand the product:
  $(14x + 3)(12x - 8) = 14x \cdot 12x + 14x \cdot (-8) + 3 \cdot 12x + 3 \cdot (-8) = 168x^2 - 112x + 36x - 24$.
• Step 6: Combine like terms:
  $168x^2 - 76x - 24$.
• Step 7: Divide by 2 to find the area:
  $A = \frac{1}{2} \times (168x^2 - 76x - 24) = 84x^2 - 38x - 12$.

Therefore, the area of the trapezoid is $84x^2 - 38x - 12$.