Alright, let's determine which of the given expressions is equivalent to the polynomial [tex]\(-5 y^2 + 50 y - 105\)[/tex].
We are given four expressions to check for equivalency:
1. [tex]\(-5(y - 3)(y - 7)\)[/tex]
2. [tex]\(-5(y + 21)(y + 1)\)[/tex]
3. [tex]\((-5 y + 21)(y - 5)\)[/tex]
4. [tex]\((5 y - 35)(-y + 7)\)[/tex]
### Step-by-Step Solution:
1. Expression 1: [tex]\(-5(y - 3)(y - 7)\)[/tex]
Expand the expression inside the parentheses:
[tex]\[
(y - 3)(y - 7) = y^2 - 7y - 3y + 21 = y^2 - 10y + 21
\][/tex]
Now, multiply by [tex]\(-5\)[/tex]:
[tex]\[
-5(y^2 - 10y + 21) = -5y^2 + 50y - 105
\][/tex]
This matches the given expression [tex]\(-5 y^2 + 50 y - 105\)[/tex].
2. Expression 2: [tex]\(-5(y + 21)(y + 1)\)[/tex]
Expand the expression inside the parentheses:
[tex]\[
(y + 21)(y + 1) = y^2 + y + 21y + 21 = y^2 + 22y + 21
\][/tex]
Now, multiply by [tex]\(-5\)[/tex]:
[tex]\[
-5(y^2 + 22y + 21) = -5y^2 - 110y - 105
\][/tex]
This does not match the given expression.
3. Expression 3: [tex]\((-5 y + 21)(y - 5)\)[/tex]
Expand the expression inside the parentheses:
[tex]\[
(-5 y + 21)(y - 5) = (-5 y)(y) + (-5 y)(-5) + (21)(y) + (21)(-5) = -5y^2 + 25y + 21y - 105 = -5y^2 + 46y - 105
\][/tex]
This does not match the given expression.
4. Expression 4: [tex]\((5 y - 35)(-y + 7)\)[/tex]
Expand the expression inside the parentheses:
[tex]\[
(5 y - 35)(-y + 7) = (5 y)(-y) + (5 y)(7) + (-35)(-y) + (-35)(7) = -5y^2 + 35y - 35y - 245 = -5y^2 - 245
\][/tex]
This does not match the given expression.
### Conclusion:
Only the first expression [tex]\(-5(y - 3)(y - 7)\)[/tex] matches the given polynomial [tex]\(-5 y^2 + 50 y - 105\)[/tex].
Therefore, the correct answer is:
[tex]\[
-5(y - 3)(y - 7)
\][/tex]
