To identify which expressions are equivalent to \( 12x^2 - 48x + 48 \), we will factor the given expression and then compare it with the provided options.
First, we can factor out the greatest common factor from \( 12x^2 - 48x + 48 \):
\[ 12x^2 - 48x + 48 = 12(x^2 - 4x + 4) \]
Next, we notice that \( x^2 - 4x + 4 \) can be factored as:
\[ x^2 - 4x + 4 = (x - 2)^2 \]
Thus, we further simplify the expression:
\[ 12(x^2 - 4x + 4) = 12(x - 2)^2 \]
Now we can check the options:
a. \( 12(x - 2)^2 \) → This is equivalent to \( 12x^2 - 48x + 48 \).
b. \( -12(x^2 + 4x + 4) \) → This can be simplified to \( -12((x + 2)^2) \) which is not equivalent to our original polynomial.
c. \( 12(x - 4)(x - 1) \) → This expands to \( 12(x^2 - 5x + 4) \) which does not match our original polynomial.
d. \( -12(x + 2)^2 \) → This expands to \( -12(x^2 + 4x + 4) \) which is not equivalent to our original polynomial.
e. \( 12(x^2 - 4x + 4) \) → This expression is equivalent to \( 12(x - 2)^2 \) which is the same as \( 12x^2 - 48x + 48 \).
Based on our checks, the two correct answers are:
a. \( 12(x - 2)^2 \)
e. \( 12(x^2 - 4x + 4) \)