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).
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\(4x^2 \cdot x^2 = 4x^4\)
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\(4x^2 \cdot (-8x) = -32x^3\)
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\(4x^2 \cdot 16 = 64x^2\)
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\(4x \cdot x^2 = 4x^3\)
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\(4x \cdot (-8x) = -32x^2\)
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\(4x \cdot 16 = 64x\)
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\(1 \cdot x^2 = x^2\)
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\(1 \cdot (-8x) = -8x\)
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\(1 \cdot 16 = 16\)
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:
\[ \boxed{4x^4 - 28x^3 + 33x^2 + 56x + 16} \]
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