The diagram shows a parallelogram - Edexcel - GCSE Maths - Question 24 - 2019 - Paper 2

To show that the inequality $2x^2 - 21x + 40 < 0$ holds true, we first need to derive the expression for the area of the parallelogram.

The area of a parallelogram can be calculated using the formula: $ext{Area} = ext{base} imes ext{height}$

Here, the base is $(2x - 1)$ cm and the height can be derived using the angle of $150^ ext{o}$: ext{height} = (10 - x) imes ext{sin}(150^ ext{o}) = (10 - x) imes rac{1}{2} = 5 - rac{x}{2}

Thus, the area becomes: ext{Area} = (2x - 1)(5 - rac{x}{2})

Expanding this expression gives: ext{Area} = 10x - x^2 - 5 + rac{x}{2} = -x^2 + 10x - 5

To ensure that the area is greater than 15 cm², we set up the inequality: $-x^2 + 10x - 5 > 15$

Rearranging yields: $-x^2 + 10x - 20 > 0$ which is equivalent to: $x^2 - 10x + 20 < 0$

However, from the original expression given in the problem $2x^2 - 21x + 40 < 0$. We can manipulate this inequality further to show consistency with our area.

Through polynomial factorization or using the quadratic formula, we find the roots of $2x^2 - 21x + 40 = 0$. The quadratic yields critical points that will help establish validity for the inequality.