Question

factor.
$4z^2 - 4z - 3$

Explanation:

Step1: Let \( y = z^2 \), rewrite the expression.

The given expression is \( 4z^2 - 4z - 3 \). Let \( y = z^2 \), then the expression becomes \( 4y^2 - 4y - 3 \) (wait, no, actually, the original expression is quadratic in \( z \), so we can factor it directly. Let's use the quadratic formula or factoring by grouping. For a quadratic \( ax^2+bx+c \), we need two numbers that multiply to \( a\times c \) and add to \( b \). Here, \( a = 4 \), \( b=-4 \), \( c = -3 \). So \( a\times c=4\times(-3)=-12 \). We need two numbers that multiply to -12 and add to -4. The numbers are -6 and 2, since \( -6\times2=-12 \) and \( -6 + 2=-4 \).

Step2: Rewrite the middle term and factor by grouping.

Rewrite \( -4z \) as \( -6z + 2z \). So the expression becomes:
\( 4z^2 - 6z + 2z - 3 \)
Group the first two and last two terms:
\( (4z^2 - 6z) + (2z - 3) \)
Factor out the greatest common factor from each group:
\( 2z(2z - 3) + 1(2z - 3) \)
Now, factor out \( (2z - 3) \):
\( (2z - 3)(2z + 1) \)

Answer:

\( (2z - 3)(2z + 1) \)