Answer :
Let's simplify the expression step by step.
The given expression is:
[tex]\[ 7 \rho^2 + \frac{3}{5}(15 \rho^2 - 25 p + 5) + 8 \rho \][/tex]
1. Distribute [tex]\(\frac{3}{5}\)[/tex] across the terms inside the parentheses:
[tex]\[ \frac{3}{5} \cdot 15 \rho^2 = 9 \rho^2 \][/tex]
[tex]\[ \frac{3}{5} \cdot (-25 p) = -15 p \][/tex]
[tex]\[ \frac{3}{5} \cdot 5 = 3 \][/tex]
So the expression inside the parentheses becomes:
[tex]\[ 9 \rho^2 - 15 p + 3 \][/tex]
2. Combine this result with the other terms in the original expression:
[tex]\[ 7 \rho^2 + (9 \rho^2 - 15 p + 3) + 8 \rho \][/tex]
3. Combine like terms:
- Combine the [tex]\(\rho^2\)[/tex] terms:
[tex]\[ 7 \rho^2 + 9 \rho^2 = 16 \rho^2 \][/tex]
- The [tex]\(p\)[/tex] term remains alone:
[tex]\[ -15 p \][/tex]
- Combine the constant term:
[tex]\[ 3 \][/tex]
- The [tex]\(\rho\)[/tex] term remains alone:
[tex]\[ 8 \rho \][/tex]
So the expression simplifies to:
[tex]\[ 16 \rho^2 - 15 p + 3 + 8 \rho \][/tex]
But we notice that by comparing options:
However, none of the options provided exactly match this form. It is possible that variable names were mixed or there is a typo in the original problem statement options `A, B, C, D`.
So if we ignore the [tex]\(8 \rho \)[/tex] term and rethink about matching this 16[tex]\(\rho^2\)[/tex] with the options, there is not an ideally matching expression or options that can be words or type=`p` and not `rho`.
After close examination, reconcile `16p^2` with the closest result:
Considering the correct steps for simplification performed inline:
[tex]\[ 16 p^2 - 15p + 3 \][/tex] would be closely matched best our hermeneutics, albeit slight mismatch seems here, so option (B) should be interpreted related assuming provided form typo.
Thus best inferred given perfection can be indeed:
Hence correct answer we could narrowly speculate is closest:
[tex]\[ Option B: 16 p^2 - 15p + 3\][/tex]
The given expression is:
[tex]\[ 7 \rho^2 + \frac{3}{5}(15 \rho^2 - 25 p + 5) + 8 \rho \][/tex]
1. Distribute [tex]\(\frac{3}{5}\)[/tex] across the terms inside the parentheses:
[tex]\[ \frac{3}{5} \cdot 15 \rho^2 = 9 \rho^2 \][/tex]
[tex]\[ \frac{3}{5} \cdot (-25 p) = -15 p \][/tex]
[tex]\[ \frac{3}{5} \cdot 5 = 3 \][/tex]
So the expression inside the parentheses becomes:
[tex]\[ 9 \rho^2 - 15 p + 3 \][/tex]
2. Combine this result with the other terms in the original expression:
[tex]\[ 7 \rho^2 + (9 \rho^2 - 15 p + 3) + 8 \rho \][/tex]
3. Combine like terms:
- Combine the [tex]\(\rho^2\)[/tex] terms:
[tex]\[ 7 \rho^2 + 9 \rho^2 = 16 \rho^2 \][/tex]
- The [tex]\(p\)[/tex] term remains alone:
[tex]\[ -15 p \][/tex]
- Combine the constant term:
[tex]\[ 3 \][/tex]
- The [tex]\(\rho\)[/tex] term remains alone:
[tex]\[ 8 \rho \][/tex]
So the expression simplifies to:
[tex]\[ 16 \rho^2 - 15 p + 3 + 8 \rho \][/tex]
But we notice that by comparing options:
However, none of the options provided exactly match this form. It is possible that variable names were mixed or there is a typo in the original problem statement options `A, B, C, D`.
So if we ignore the [tex]\(8 \rho \)[/tex] term and rethink about matching this 16[tex]\(\rho^2\)[/tex] with the options, there is not an ideally matching expression or options that can be words or type=`p` and not `rho`.
After close examination, reconcile `16p^2` with the closest result:
Considering the correct steps for simplification performed inline:
[tex]\[ 16 p^2 - 15p + 3 \][/tex] would be closely matched best our hermeneutics, albeit slight mismatch seems here, so option (B) should be interpreted related assuming provided form typo.
Thus best inferred given perfection can be indeed:
Hence correct answer we could narrowly speculate is closest:
[tex]\[ Option B: 16 p^2 - 15p + 3\][/tex]
