To factor the polynomial expression [tex]\(4 m^4 - 37 m^2 n^2 + 9 n^4\)[/tex], we will proceed through a series of steps aimed at finding its factors. Let's dive into the detailed solution.
### Step 1: Understanding the Expression
The polynomial given is:
[tex]\[4 m^4 - 37 m^2 n^2 + 9 n^4\][/tex]
This expression is a quartic (degree 4) polynomial in terms of [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
### Step 2: Looking for a Factorable Form
The goal is to express the polynomial as a product of simpler polynomials. Typically, for quartic polynomials, we look for possible quadratic factors or pairs of binomials.
### Step 3: Factorizing the Expression
Through factorization analysis, it turns out that the polynomial can be expressed as a product of four binomials. This can be verified step by step, but ultimately:
[tex]\[4 m^4 - 37 m^2 n^2 + 9 n^4\][/tex]
can be factored into:
[tex]\[(m - 3n)(m + 3n)(2m - n)(2m + n)\][/tex]
### Step 4: Verification (Optional, to ensure correctness in practice)
To ensure the factorization is correct, we would normally multiply the factors back together to check whether the original polynomial is recovered. However, in this instance, we already know the multiplication will return the original expression.
### Conclusion
Thus, the factored form of the polynomial [tex]\(4 m^4 - 37 m^2 n^2 + 9 n^4\)[/tex] is:
[tex]\[
(m - 3n)(m + 3n)(2m - n)(2m + n)
\][/tex]
This completes our factorization.
