This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the factorization problems: 2.2 If we group the first two terms and the last two terms as follows: (xy + 5y) + (2x + 10) what do you notice about each group? Each group has a common factor. In Group 1, y is common. In Group 2, 2 is common. 2.3 Factor these values out of each group and then write down the equivalent algebraic expression. Step 1: Factor Group 1. xy + 5y = y(x + 5) Step 2: Factor Group 2. 2x + 10 = 2(x + 5) Step 3: Write the equivalent algebraic expression. y(x + 5) + 2(x + 5) The equivalent algebraic expression is y(x + 5) + 2(x + 5). 2.4 What is the common factor in the two terms? The common factor in the two terms y(x + 5) and 2(x + 5) is (x + 5). 2.5 Use the distributive property to factor out this common factor and then express the polynomial as a product of two binomials. Step 1: Factor out the common binomial factor (x + 5). y(x + 5) + 2(x + 5) = (x + 5)(y + 2) The polynomial as a product of two binomials is (x + 5)(y + 2). 2.6 Joseph wants to factorize the following algebraic expression: 3x^2 + 6x + 4x + 8 Provide Joseph with a three-step guide on how to factorize the expression. Step 1: Group the terms into two pairs. (3x^2 + 6x) + (4x + 8) Step 2: Factor out the greatest common factor (GCF) from each group. From the first group, 3x^2 + 6x, the GCF is 3x, so it becomes 3x(x + 2). From the second group, 4x + 8, the GCF is 4, so it becomes 4(x + 2). The expression is now 3x(x + 2) + 4(x + 2). Step 3: Factor out the common binomial factor. The common binomial factor is (x + 2). (x + 2)(3x + 4) The factored expression is (x + 2)(3x + 4). 3 done, 2 left today. You're making progress.