Question

or
Find the product. Simplify your answer.
(4r–4)(4r+2)

Answer 1

To find the product of the expression \((4r - 4)(4r + 2)\), we can use the distributive property (also known as the FOIL method for binomials).

Here’s how it works step by step:

1. **Distribute the first term from the first binomial:**
\[
4r \cdot (4r + 2) = 4r \cdot 4r + 4r \cdot 2 = 16r^2 + 8r
\]

2. **Distribute the second term from the first binomial:**
\[
-4 \cdot (4r + 2) = -4 \cdot 4r - 4 \cdot 2 = -16r - 8
\]

3. **Combine all the terms:**
\[
16r^2 + 8r - 16r - 8
\]

4. **Simplify by combining like terms:**
\[
16r^2 + (8r - 16r) - 8 = 16r^2 - 8r - 8
\]

5. **Factor out the common factor:**
\[
16r^2 - 8r - 8 = 8(2r^2 - r - 1)
\]

To factor \(2r^2 - r - 1\), we look for two numbers that multiply to \(-2\) (the product of the coefficient of \(r^2\) times the constant term) and add to \(-1\) (the coefficient of \(r\)). These numbers are \(-2\) and \(1\).

We can rewrite the quadratic:
\[
2r^2 - 2r + r - 1 = 2r(r - 1) + 1(r - 1) = (2r + 1)(r - 1)
\]

So, the final factored form of the entire expression is:
\[
8(2r + 1)(r - 1)
\]

Thus, the simplified answer to the product \((4r - 4)(4r + 2)\) is:
\[
\boxed{8(2r + 1)(r - 1)}
\]