Factorize the Quadratic x² - 3x - 4: Understanding Product Representation

To solve this problem, let's follow these steps:

• Step 1: Identify the coefficients: $a = 1$, $b = -3$, $c = -4$.
• Step 2: Calculate the product $ac = 1 \times -4 = -4$ and the sum $b = -3$.
• Step 3: Find two numbers that multiply to $-4$ and add to $-3$. These numbers are $-4$ and $1$.
• Step 4: Rewrite the middle term $-3x$ using $-4$ and $1$ as $-4x + x$.
• Step 5: Factor by grouping: $(x^2 - 4x) + (x - 4)$.
• Step 6: Factor out common factors: $x(x - 4) + 1(x - 4)$.
• Step 7: Notice the common binomial factor: $(x - 4)(x + 1)$.

Therefore, the factored form of the quadratic function $f(x) = x^2 - 3x - 4$ is $(x - 4)(x + 1)$, which corresponds to choice 3.