To determine which expression is equivalent to [tex]\(x^2 - 12x + 11\)[/tex], we need to factor the quadratic expression.
Factoring a quadratic expression [tex]\(ax^2 + bx + c\)[/tex] involves finding two binomials [tex]\((x - p)(x - q)\)[/tex] such that when expanded, they yield the original quadratic expression.
Given the quadratic expression:
[tex]\[ x^2 - 12x + 11 \][/tex]
We need two numbers that multiply to the constant term [tex]\(c\)[/tex] (which is 11) and add to the coefficient of the linear term [tex]\(b\)[/tex] (which is -12).
The numbers that satisfy these conditions are -11 and -1.
Thus, we can write the factored form of the quadratic expression as:
[tex]\[ (x - 11)(x - 1) \][/tex]
To verify, we can expand [tex]\((x - 11)(x - 1)\)[/tex] to check if it matches the original quadratic expression:
[tex]\[
(x - 11)(x - 1) = x^2 - x - 11x + 11 = x^2 - 12x + 11
\][/tex]
The expanded form [tex]\(x^2 - 12x + 11\)[/tex] matches the original quadratic expression.
Therefore, the expression equivalent to [tex]\(x^2 - 12x + 11\)[/tex] is:
[tex]\[
\boxed{(x - 11)(x - 1)}
\][/tex]
