Find the product of the polynomials (2x+1)(x−4)(2x+1)(x−4) .(2 points) x2+x2+x+x+

To find the product of the polynomials \((2x+1)(x−4)(2x+1)(x−4)\), we can first recognize that this is the same as \(((2x + 1)(x - 4))^2\).

### Step 1: Expand \((2x + 1)(x - 4)\)

We can use the distributive property (also known as the FOIL method for binomials):

\[
(2x + 1)(x - 4) = 2x \cdot x + 2x \cdot (-4) + 1 \cdot x + 1 \cdot (-4)
\]

Calculating each term, we get:

\[
= 2x^2 - 8x + x - 4
\]
\[
= 2x^2 - 7x - 4
\]

### Step 2: Square the result

Now we need to square \((2x^2 - 7x - 4)\):

\[
(2x^2 - 7x - 4)^2
\]

Using the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\), where \(a = 2x^2\), \(b = -7x\), and \(c = -4\):

1. \(a^2 = (2x^2)^2 = 4x^4\)
2. \(b^2 = (-7x)^2 = 49x^2\)
3. \(c^2 = (-4)^2 = 16\)

Now we calculate the cross terms:

- \(2ab = 2(2x^2)(-7x) = -28x^3\)
- \(2ac = 2(2x^2)(-4) = -16x^2\)
- \(2bc = 2(-7x)(-4) = 56x\)

Now adding everything together:

\[
4x^4 - 28x^3 + (49x^2 - 16x^2) + 56x + 16
\]

Simplifying the \(x^2\) term:

\[
= 4x^4 - 28x^3 + 33x^2 + 56x + 16
\]

### Final Result

Thus, the product of the polynomials \((2x+1)(x-4)(2x+1)(x-4)\) is:

\[
\boxed{4x^4 - 28x^3 + 33x^2 + 56x + 16}
\]