Select the correct answer.

Which expression is equivalent to the given expression?

[tex]-5y^2 + 50y - 105[/tex]

A. [tex]-5(y - 3)(y - 7)[/tex]
B. [tex]-5(y + 21)(y + 1)[/tex]
C. [tex](-5y + 21)(y - 5)[/tex]
D. [tex](5y - 35)(-y + 7)[/tex]



Answer :

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]