Simplify the Expression: 7z+10b+2bz+35 Using Like Terms

Which of the expressions is equivalent to the expression?

$7z+10b+2bz+35$

To solve this problem, we'll focus on factorization by grouping:

• Step 1: Identify groupable terms in $7z + 10b + 2bz + 35$.
• Step 2: Reorganize and group terms for greatest common factor extraction.
• Step 3: Factor each group and simplify.

Now, let's work through each step:

Step 1:
Observe that we can reorganize the expression to facilitate grouping:

$7z + 35 + 10b + 2bz$.

Step 2:
Group into pairs: $(7z + 35) + (10b + 2bz)$.
Within each pair, extract common factors:

$= 7(z + 5) + 2b(z + 5)$, noticing that each group factors nicely.

Step 3:
Since both terms now have a common factor of $(z + 5)$, we can factor it out:

$= (7 + 2b)(z + 5)$.

Therefore, the expression $7z + 10b + 2bz + 35$ is equivalent to $(7 + 2b)(z + 5)$.

This matches choice 1: $(7+2b)(z+5)$.