To solve this problem of factoring the expression [tex]\( 10h + 30 + 50j \)[/tex], let's consider each of the given options step by step to identify which ones are equivalent to the original expression.
### Original Expression:
[tex]\[ 10h + 30 + 50j \][/tex]
### Factoring Step-by-Step:
1. Option (A): [tex]\( 5(2h + 30 + 50j) \)[/tex]:
- Let's distribute the 5:
[tex]\[
5(2h + 30 + 50j) = 5 \cdot 2h + 5 \cdot 30 + 5 \cdot 50j = 10h + 150 + 250j
\][/tex]
- Compare this with the original expression [tex]\( 10h + 30 + 50j \)[/tex]:
[tex]\[
10h + 150 + 250j \neq 10h + 30 + 50j
\][/tex]
- Therefore, Option (A) is not equivalent to [tex]\( 10h + 30 + 50j \)[/tex].
2. Option (B): [tex]\( 10(h + 3 + 5j) \)[/tex]:
- Let's distribute the 10:
[tex]\[
10(h + 3 + 5j) = 10 \cdot h + 10 \cdot 3 + 10 \cdot 5j = 10h + 30 + 50j
\][/tex]
- Compare this with the original expression [tex]\( 10h + 30 + 50j \)[/tex]:
[tex]\[
10h + 30 + 50j = 10h + 30 + 50j
\][/tex]
- Therefore, Option (B) is equivalent to [tex]\( 10h + 30 + 50j \)[/tex].
3. Option (C): [tex]\( 2(5h + 15 + 25j) \)[/tex]:
- Let's distribute the 2:
[tex]\[
2(5h + 15 + 25j) = 2 \cdot 5h + 2 \cdot 15 + 2 \cdot 25j = 10h + 30 + 50j
\][/tex]
- Compare this with the original expression [tex]\( 10h + 30 + 50j \)[/tex]:
[tex]\[
10h + 30 + 50j = 10h + 30 + 50j
\][/tex]
- Therefore, Option (C) is equivalent to [tex]\( 10h + 30 + 50j \)[/tex].
4. Option (D): [tex]\( 15(10h + 2 + 50j) \)[/tex]:
- Let's distribute the 15:
[tex]\[
15(10h + 2 + 50j) = 15 \cdot 10h + 15 \cdot 2 + 15 \cdot 50j = 150h + 30 + 750j
\][/tex]
- Compare this with the original expression [tex]\( 10h + 30 + 50j \)[/tex]:
[tex]\[
150h + 30 + 750j \neq 10h + 30 + 50j
\][/tex]
- Therefore, Option (D) is not equivalent to [tex]\( 10h + 30 + 50j \)[/tex].
### Conclusion:
After the step-by-step examination, we find that:
- Option (B) [tex]\( 10(h + 3 + 5j) \)[/tex] is correct.
- Option (C) [tex]\( 2(5h + 15 + 25j) \)[/tex] is correct.
Thus, the two equivalent expressions to [tex]\( 10h + 30 + 50j \)[/tex] are:
- \[ B \) [tex]\( 10(h + 3 + 5j) \)[/tex]]
- \[ C \) [tex]\( 2(5h + 15 + 25j) \)[/tex]]
