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A quadratic equation is a type of polynomial equation of degree 2. It typically has the form: y = ax^2 + bx + c,where a, b, and c are constants and x represents the variable. In this specific problem, the original polynomial expression is written as 16x^4 + 16x^2 + 3. It might look more complex, but it follows a similar pattern. Once we recognize that 16x^4 = (4x^2)^2,we can rewrite the polynomial in terms of u = x^2,transforming the expression into a standard quadratic form 16u^2 + 16u + 3.Now, it looks much simpler and we can use our understanding of quadratic equations to solve it.

The substitution method is a useful strategy in algebra for simplifying and solving equations by replacing variables with other variables or expressions. In our problem, we start with a more complex trinomial, 16x^4 + 16x^2 + 3. To simplify it, we make a substitution:u = x^2.This changes the original expression into a simpler quadratic form:16u^2 + 16u + 3.By performing this substitution, we reduce a polynomial of degree 4 to a quadratic equation of degree 2. After factoring the quadratic equation, we reverse the substitution to return to the original variable x. This method helps to break down complex problems into more manageable parts.

Polynomial factorization involves expressing a polynomial as a product of its factors. This is crucial for solving polynomial equations and simplifying expressions. In our exercise, after making the substitution u = x^2, we factorize the transformed quadratic expression 16u^2 + 16u + 3.We find its factors by finding two binomials that multiply to produce the quadratic:(4u + 3)(4u + 1).Once we factorize the quadratic in terms of u, we revert to the original variable x. By substituting back u = x^2, we obtain the final factored form of the original polynomial (4x^2 + 3)(4x^2 + 1).This factorization provides a simpler form for the polynomial, which can be beneficial for further algebraic manipulations or solving equations.