The Correct Answer and Explanation is:
The image displays the mathematical expression: 5x² + 32x + 35.
This expression is a quadratic trinomial. It is “quadratic” because the highest power of the variable x is 2 (the x² term). It is a “trinomial” because it consists of three distinct terms: 5x², 32x, and 35, all connected by addition. In the standard form of a quadratic, ax² + bx + c, the coefficients for this specific expression are a = 5, b = 32, and c = 35.
A common operation performed on such expressions is factoring, which involves rewriting it as a product of two or more simpler expressions, typically linear binomials. To factor 5x² + 32x + 35, one can use the “ac method”. This involves finding two numbers that multiply to the product of a and c (5 × 35 = 175) and add up to b (32). The factors of 175 are pairs like (1, 175), (5, 35), and (7, 25). The pair (7, 25) is the correct choice because 7 + 25 = 32.
Using this pair, we can rewrite the middle term, 32x, as 7x + 25x. The expression becomes 5x² + 25x + 7x + 35. Now, we can factor by grouping. We group the first two terms and the last two terms: (5x² + 25x) + (7x + 35). From the first group, we factor out the greatest common factor, 5x, to get 5x(x + 5). From the second group, we factor out 7 to get 7(x + 5). Since both parts share the common binomial factor (x + 5), we can complete the factoring process.
The final factored form of the expression is (5x + 7)(x + 5). This factored form is equivalent to the original expression and is useful for solving quadratic equations or analyzing the graph of the corresponding function.
