To find the expression equivalent to [tex]\((p - q)(x)\)[/tex], we need to calculate [tex]\(p(x) - q(x)\)[/tex].
First, let's rewrite the given functions for clarity:
[tex]\[ p(x) = x^2 - 1 \][/tex]
[tex]\[ q(x) = 5(x - 1) \][/tex]
Now, we will calculate [tex]\(p(x) - q(x)\)[/tex]:
[tex]\[ (p - q)(x) = p(x) - q(x) \][/tex]
Substitute the functions [tex]\(p(x)\)[/tex] and [tex]\(q(x)\)[/tex]:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5(x - 1) \][/tex]
Next, simplify the expression:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5x + 5 \][/tex]
Combining like terms, we get:
[tex]\[ (p - q)(x) = x^2 - 1 - 5x + 5 \][/tex]
[tex]\[ (p - q)(x) = x^2 - 5x - 1 + 5 \][/tex]
[tex]\[ (p - q)(x) = x^2 - 5x + 4 \][/tex]
Therefore, the correct expression equivalent to [tex]\((p - q)(x)\)[/tex] is:
[tex]\[ \left(x^2 - 1\right) - 5(x - 1) \][/tex]
This matches the third option in the list provided:
[tex]\[ \left(x^2 - 1\right) - 5(x - 1) \][/tex]
