Match Equivalent Expressions: (2a+b)(b+4) and Related Polynomials

To solve this problem, we need to expand each given expression and compare it to the list of provided expanded expressions.

Let's expand each of the four expressions:

• Expression 1: $(2a+b)(b+4)$

Applying the distributive property:

$(2a+b)(b) + (2a+b)(4) = 2ab + b^2 + 8a + 4b$ This matches the expanded form $2ab + b^2 + 8a + 4b$, option d.
• Expression 2: $(4+a)(2b+b) = (4+a)(3b)$

Distributing the terms:

$3b \cdot 4 + 3b \cdot a = 12b + 3ab$ This matches the expanded form $12b + 3ab$, option b.
• Expression 3: $(2a-b)(b-4)$

Distributing the terms:

$2a \cdot b - 2a \cdot 4 - b \cdot b + b \cdot 4 = 2ab - 8a - b^2 + 4b$ This matches the expanded form $2ab - 8a - b^2 + 4b$, option a.
• Expression 4: $(2a-b)(b+4)$

Distributing the terms:

$2a \cdot b + 2a \cdot 4 - b \cdot b - b \cdot 4 = 2ab + 8a - b^2 - 4b$ This matches the expanded form $2ab + 8a - b^2 - 4b$, option c.

Therefore, the correct matching is as follows:

• Expression 1: $(2a+b)(b+4)$ matches with $2ab+8a+b^2+4b$
• Expression 2: $(4+a)(2b+b)$ matches with $12b+3ab$
• Expression 3: $(2a-b)(b-4)$ matches with $2ab-8a-b^2+4b$
• Expression 4: $(2a-b)(b+4)$ matches with $2ab+8a-b^2-4b$

Thus, the correct answer is: 1-d, 2-b, 3-a, 4-c.