Factoring the Quadratic Function: Find the Product of x²+11x+28

To solve this problem, we need to find a product representation of the quadratic function $f(x) = x^2 + 11x + 28$.

Let's go through the problem-solving process step-by-step:

1. We need to express the given quadratic function $f(x) = x^2 + 11x + 28$ as a product of two binomials. 2. Key information includes the coefficients of the terms in the quadratic: 1 for $x^2$, 11 for $x$, and 28 for the constant term. 3. The approach is to factor the quadratic expression. 4. Key formula: For a quadratic expression $ax^2 + bx + c$, the product of factors $(x+p)(x+q)$ requires that $p+q = b/a$ and $pq = c/a$. 5. Factoring by finding two numbers whose sum is 11 and product is 28. 6. Steps: Find two numbers that satisfy the conditions $p+q = 11$ and $pq = 28$. 7. Assumptions: The quadratic can be factored into real numbers. 8. Consider pairs of factors of 28: $(1, 28), (2, 14), (4, 7), (-1, -28), (-2, -14), (-4, -7)$. 9. Edge cases: Check that potential factors satisfy both conditions. 10. We consider only the choices since the problem is a multiple-choice. 11. For multiple-choice questions, find the correct pair of numbers by testing the conditions. 12. Common mistakes might include not correctly identifying factor pairs or choosing incorrect signs. 13. Changes in coefficients would alter the factor pairs needed to produce the correct sum and product.

Now let's address the problem:

To find the correct binomial factors of $f(x) = x^2 + 11x + 28$, we are searching for two numbers that multiply to 28 and add to 11. Examining possible pairs, $4$ and $7$ meet these criteria: $4 \times 7 = 28$ and $4 + 7 = 11$.

Thus, the quadratic expression can be rewritten as:

$f(x) = (x + 4)(x + 7)$

This checks our desired conditions. Multiplication confirms: $(x+4)(x+7) = x^2 + 7x + 4x + 28 = x^2 + 11x + 28$, which matches the original function.

Therefore, the product representation of $f(x) = x^2 + 11x + 28$ is $(x+7)(x+4)$.