Find each product. (x+1)(x+1)(x-1)(x-1)

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Recognize that the expression is a product of two pairs of binomials: $(x+1)(x+1)$ and $(x-1)(x-1)$.

Rewrite each pair as a square: $(x+1)^2$ and $(x-1)^2$.

Recall the formula for the difference of squares: $(a+b)(a-b) = a^2 - b^2$. Here, we have $(x+1)^2 (x-1)^2$, which can be seen as $[(x+1)(x-1)]^2$.

Apply the difference of squares to $(x+1)(x-1)$ to get $x^2 - 1$.

Square the result from the previous step to get $(x^2 - 1)^2$, which is the product of the original expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomial expressions by applying the distributive property. Each term in one polynomial is multiplied by every term in the other, and like terms are combined to simplify the result.

Difference of Squares

The difference of squares is a special product formula: (a + b)(a - b) = a² - b². Recognizing this pattern helps simplify expressions quickly without full expansion, especially when dealing with conjugate binomials.

Combining Like Terms

After multiplying polynomials, like terms (terms with the same variable raised to the same power) must be combined by adding or subtracting their coefficients. This step simplifies the expression into its standard polynomial form.

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