Represent the Product of the Quadratic Function: x²-5x-50

To solve this problem, we will factor the quadratic function $f(x) = x^2 - 5x - 50$ into two binomials:

• Step 1: Identify the values of $a = 1$, $b = -5$, and $c = -50$ in the quadratic expression $ax^2 + bx + c$.
• Step 2: We look for two numbers that multiply to $c = -50$ and add up to $b = -5$.
• Step 3: Consider the factor pairs of $-50$. Possible pairs include $(-1, 50)$, $(1, -50)$, $(-2, 25)$, $(2, -25)$, $(-5, 10)$, and $(5, -10)$.
• Step 4: The correct pair is $(5, -10)$ because $5 \times (-10) = -50$ and $5 + (-10) = -5$.
• Step 5: Express $f(x)$ in its factored form using these numbers: $f(x) = (x + 5)(x - 10)$.
• Step 6: Verify the factorization by expanding: $(x + 5)(x - 10) = x^2 - 10x + 5x - 50$, which simplifies to $x^2 - 5x - 50$, confirming correctness.

Therefore, the correct factorization of the quadratic $f(x) = x^2 - 5x - 50$ is $(x + 5)(x - 10)$.

Thus, the product representation of the function is $(x+5)(x-10)$.