Derive the Product Form of the Quadratic f(x) = x² + x - 2

Find the representation of the product of the following function

$f(x)=x^2+x-2$

To determine the product representation of $f(x) = x^2 + x - 2$, we can factor the quadratic equation by following these steps:

• Step 1: Identify the product $ac = 1 \times (-2) = -2$ and sum $b = 1$.
• Step 2: Find two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1$.
• Step 3: Rewrite the middle term using these numbers: $x^2 + 2x - 1x - 2$.
• Step 4: Factor by grouping:
  - Group $x^2 + 2x$ and $-1x - 2$ as separate pairs:
  - $x(x + 2) - 1(x + 2)$.
• Step 5: Factor out the common terms:
  $(x + 2)(x - 1)$.

Thus, the product representation of the function is $(x + 2)(x - 1)$.