To determine which expression is equivalent to the given expression [tex]\( -5y^2 + 50y - 105 \)[/tex], we need to factor the quadratic expression. Here's a detailed, step-by-step solution:
1. Identify the given expression:
[tex]\[ -5y^2 + 50y - 105 \][/tex]
2. Factor out the greatest common factor:
The common factor in all terms is [tex]\(-5\)[/tex]. So, factor [tex]\(-5\)[/tex] out of the expression:
[tex]\[ -5(y^2 - 10y + 21) \][/tex]
3. Factor the quadratic expression:
Next, we need to factor the quadratic expression [tex]\( y^2 - 10y + 21 \)[/tex]. We look for two numbers that add up to [tex]\(-10\)[/tex] and multiply to [tex]\(21\)[/tex]:
These numbers are [tex]\(-3\)[/tex] and [tex]\(-7\)[/tex], because:
- [tex]\(-3 + (-7) = -10\)[/tex]
- [tex]\(-3 \times (-7) = 21\)[/tex]
4. Rewrite and factor the expression:
Using the numbers we found, we can rewrite the expression inside the parenthesis:
[tex]\[ y^2 - 10y + 21 = (y - 3)(y - 7) \][/tex]
5. Combine the factors:
Now place the factored quadratic back into the expression with [tex]\(-5\)[/tex] factored out:
[tex]\[ -5(y - 3)(y - 7) \][/tex]
Thus, the equivalent expression to [tex]\( -5y^2 + 50y - 105 \)[/tex] is:
[tex]\[ -5(y - 3)(y - 7) \][/tex]
So, the correct choice from the given options is:
[tex]\[ -5(y-3)(y-7) \][/tex]
