Answer :
We are given the functions
[tex]$$
f(x) = -12x^4 + 5x^2 - 45
$$[/tex]
and
[tex]$$
g(x) = 10x^4 - 70x^2 - 300.
$$[/tex]
To find [tex]$(f - g)(x)$[/tex], we subtract [tex]$g(x)$[/tex] from [tex]$f(x)$[/tex]:
[tex]$$
(f - g)(x) = f(x) - g(x).
$$[/tex]
Substitute the expressions for [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]:
[tex]$$
(f - g)(x) = \left(-12x^4 + 5x^2 - 45\right) - \left(10x^4 - 70x^2 - 300\right).
$$[/tex]
Next, remove the parentheses. Be careful with the negative sign before the second set of parentheses:
[tex]$$
(f - g)(x) = -12x^4 + 5x^2 - 45 - 10x^4 + 70x^2 + 300.
$$[/tex]
Now, combine like terms starting with the [tex]$x^4$[/tex] terms, then the [tex]$x^2$[/tex] terms, and finally the constant terms:
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$
-12x^4 - 10x^4 = -22x^4.
$$[/tex]
2. Combine the [tex]$x^2$[/tex] terms:
[tex]$$
5x^2 + 70x^2 = 75x^2.
$$[/tex]
3. Combine the constant terms:
[tex]$$
-45 + 300 = 255.
$$[/tex]
Thus, the simplified form is
[tex]$$
(f - g)(x) = -22x^4 + 75x^2 + 255.
$$[/tex]
So the correct answer is
[tex]$$
-22x^4 + 75x^2 + 255.
$$[/tex]
[tex]$$
f(x) = -12x^4 + 5x^2 - 45
$$[/tex]
and
[tex]$$
g(x) = 10x^4 - 70x^2 - 300.
$$[/tex]
To find [tex]$(f - g)(x)$[/tex], we subtract [tex]$g(x)$[/tex] from [tex]$f(x)$[/tex]:
[tex]$$
(f - g)(x) = f(x) - g(x).
$$[/tex]
Substitute the expressions for [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]:
[tex]$$
(f - g)(x) = \left(-12x^4 + 5x^2 - 45\right) - \left(10x^4 - 70x^2 - 300\right).
$$[/tex]
Next, remove the parentheses. Be careful with the negative sign before the second set of parentheses:
[tex]$$
(f - g)(x) = -12x^4 + 5x^2 - 45 - 10x^4 + 70x^2 + 300.
$$[/tex]
Now, combine like terms starting with the [tex]$x^4$[/tex] terms, then the [tex]$x^2$[/tex] terms, and finally the constant terms:
1. Combine the [tex]$x^4$[/tex] terms:
[tex]$$
-12x^4 - 10x^4 = -22x^4.
$$[/tex]
2. Combine the [tex]$x^2$[/tex] terms:
[tex]$$
5x^2 + 70x^2 = 75x^2.
$$[/tex]
3. Combine the constant terms:
[tex]$$
-45 + 300 = 255.
$$[/tex]
Thus, the simplified form is
[tex]$$
(f - g)(x) = -22x^4 + 75x^2 + 255.
$$[/tex]
So the correct answer is
[tex]$$
-22x^4 + 75x^2 + 255.
$$[/tex]
