To determine which expression is equivalent to the given polynomial expression, let's simplify the given expression step by step:
Given polynomial expression:
[tex]\[ 13p - 3p(1 - 2m) - m(2m - 5p) - m^2 \][/tex]
1. Distribute [tex]\( -3p \)[/tex] inside the first parenthesis:
[tex]\[ -3p(1 - 2m) = -3p + 6pm \][/tex]
2. Distribute [tex]\(m\)[/tex] inside the second parenthesis:
[tex]\[ -m(2m - 5p) = -2m^2 + 5mp \][/tex]
Now, the expression becomes:
[tex]\[ 13p - 3p + 6pm - 2m^2 + 5mp - m^2 \][/tex]
3. Combine like terms:
[tex]\[ 13p - 3p = 10p \][/tex]
For the [tex]\( mp \)[/tex] terms:
[tex]\[ 6mp + 5mp = 11mp \][/tex]
And for the [tex]\( m^2 \)[/tex] terms:
[tex]\[ -2m^2 - m^2 = -3m^2 \][/tex]
Putting it all together:
[tex]\[ 10p + 11mp - 3m^2 \][/tex]
Now we have the simplified expression:
[tex]\[ 10p + 11mp - 3m^2 \][/tex]
Comparing this simplified expression with the choices given:
1. [tex]\(-3m^2 - 15mp + 10p\)[/tex]
2. [tex]\(-3m^2 + 11mp + 10p\)[/tex]
3. [tex]\(-3m^2 - 11mp + 10p\)[/tex]
It is clear that the simplified expression [tex]\[ 10p + 11mp - 3m^2 \][/tex] is equivalent to the second choice:
[tex]\[ -3m^2 + 11mp + 10p \][/tex]
Therefore, the expression equivalent to the given polynomial expression is:
[tex]\[ -3m^2 + 11mp + 10p \][/tex]
