To determine which function [tex]\( h(x) \)[/tex] satisfies the conditions [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex], we will evaluate each of the given functions at [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex].
Let's consider each function and evaluate accordingly:
1. [tex]\( h(x) = x^2 - 13x - 30 \)[/tex]
[tex]\[
h(3) = 3^2 - 13(3) - 30 = 9 - 39 - 30 = -60
\][/tex]
[tex]\[
h(-10) = (-10)^2 - 13(-10) - 30 = 100 + 130 - 30 = 200
\][/tex]
2. [tex]\( h(x) = x^2 - 7x - 30 \)[/tex]
[tex]\[
h(3) = 3^2 - 7(3) - 30 = 9 - 21 - 30 = -42
\][/tex]
[tex]\[
h(-10) = (-10)^2 - 7(-10) - 30 = 100 + 70 - 30 = 140
\][/tex]
3. [tex]\( h(x) = 2x^2 + 26x - 60 \)[/tex]
[tex]\[
h(3) = 2(3^2) + 26(3) - 60 = 2(9) + 78 - 60 = 18 + 78 - 60 = 36
\][/tex]
[tex]\[
h(-10) = 2(-10)^2 + 26(-10) - 60 = 2(100) - 260 - 60 = 200 - 260 - 60 = -120
\][/tex]
4. [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex]
[tex]\[
h(3) = 2(3^2) + 14(3) - 60 = 2(9) + 42 - 60 = 18 + 42 - 60 = 0
\][/tex]
[tex]\[
h(-10) = 2(-10)^2 + 14(-10) - 60 = 2(100) - 140 - 60 = 200 - 140 - 60 = 0
\][/tex]
From our evaluations:
- For [tex]\( h(x) = x^2 - 13x - 30 \)[/tex], [tex]\( h(3) = -60 \)[/tex] and [tex]\( h(-10) = 200 \)[/tex].
- For [tex]\( h(x) = x^2 - 7x - 30 \)[/tex], [tex]\( h(3) = -42 \)[/tex] and [tex]\( h(-10) = 140 \)[/tex].
- For [tex]\( h(x) = 2x^2 + 26x - 60 \)[/tex], [tex]\( h(3) = 36 \)[/tex] and [tex]\( h(-10) = -120 \)[/tex].
- For [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex], [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex].
Only the function [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex] satisfies both conditions [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex].
Thus, the function [tex]\( h(x) \)[/tex] that represents [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = 2x^2 + 14x - 60
\][/tex]
