Question

Melody is factoring the expression 13 - 2:2 + 41 – 8. Her first two steps are shown. 3 - 2x² + 4x - 8 = x²(x - 2) + 4(x - 2) (x + 4)(x - 2) What should Melody do next to finish factoring the expression? A. Melody should rewrite (x² + 4) as (x - 2) (x + 2). B. Melody has completely factored the expression. C. Melody should rewrite (x² + 4) as (x+ 2)². D. Melody should rewrite (x² + 4) as (x - 2)².​

Answer 1

To finish factoring the expression, Melody should rewrite (x + 4)(x - 2) as x^2 + 2x - 8.

Step-by-step explanation:

To finish factoring the expression 13 - 2:2 + 41 – 8, Melody needs to simplify (x + 4)(x - 2). In the expression (x + 4)(x - 2), Melody can use the FOIL method to expand it into x^2 - 2x + 4x - 8. This simplifies to x^2 + 2x - 8. So, Melody should rewrite (x + 4)(x - 2) as x^2 + 2x - 8.

Answer 2

Melody has correctly factored the expression into (x² + 4)(x - 2), and this represents the fully factored form over the real numbers, as (x² + 4) cannot be factored further with real numbers.

Step-by-step explanation:

The problem involves factoring a quadratic expression. Melody has factored the expression to x²(x - 2) + 4(x - 2), which can be factored further by grouping. Melody recognizes that both terms share a common binomial factor of (x - 2). So, the expression can be factored as (x² + 4)(x - 2). However, the expression (x² + 4) cannot be factored further over the real numbers because it does not have real roots; it is not a difference of squares and does not factor into real linear factors. Therefore, the correct next step for Melody is to conclude that she has completely factored the expression, which corresponds to option B.