Question

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

Answer 1

To find the product of the polynomials \((2x+1)(x-4)(2x+1)(x-4)\), we can first simplify this expression by rewriting it:

\[
(2x+1)^2 (x-4)^2
\]

Now, we will compute the squares of each polynomial separately.

**Step 1: Square \(2x+1\)**

\[
(2x + 1)^2 = (2x)^2 + 2(2x)(1) + (1)^2 = 4x^2 + 4x + 1
\]

**Step 2: Square \(x-4\)**

\[
(x - 4)^2 = x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16
\]

**Step 3: Now, we multiply the two results together.**

We need to compute:

\[
(4x^2 + 4x + 1)(x^2 - 8x + 16)
\]

We will use the distributive property (also known as the FOIL method for binomials).

1. \(4x^2 \cdot x^2 = 4x^4\)
2. \(4x^2 \cdot (-8x) = -32x^3\)
3. \(4x^2 \cdot 16 = 64x^2\)

4. \(4x \cdot x^2 = 4x^3\)
5. \(4x \cdot (-8x) = -32x^2\)
6. \(4x \cdot 16 = 64x\)

7. \(1 \cdot x^2 = x^2\)
8. \(1 \cdot (-8x) = -8x\)
9. \(1 \cdot 16 = 16\)

**Step 4: Combine all the terms:**

Now combine all the like terms from the above results:

\[
4x^4 + (-32x^3 + 4x^3) + (64x^2 - 32x^2 + x^2) + (64x - 8x) + 16
\]

This simplifies to:

\[
4x^4 - 28x^3 + (64 - 32 + 1)x^2 + (64 - 8)x + 16
\]

Calculating the coefficients:

- The \(x^3\) term: \(-28x^3\)
- The \(x^2\) term: \(33x^2\)
- The \(x\) term: \(56x\)

Therefore, the final result is:

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

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

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