Certainly! Let's solve the expression [tex]\(4x^2y^2 + 28xy + 49\)[/tex] by factoring it step-by-step.
1. Identify the quadratic form:
The given expression is in the form of a quadratic polynomial in terms of [tex]\(xy\)[/tex]:
[tex]\[
4x^2y^2 + 28xy + 49
\][/tex]
2. Recognize perfect square trinomial:
Notice that the expression fits the pattern of a perfect square trinomial. A perfect square trinomial has the form [tex]\((ax + b)^2 = a^2x^2 + 2abx + b^2\)[/tex].
3. Match the form:
To factor the expression, we need to find values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[
(2xy + 7)^2 = (2xy)^2 + 2 \cdot 2xy \cdot 7 + 7^2
\][/tex]
4. Verification:
Let's expand [tex]\((2xy + 7)^2\)[/tex] step-by-step to verify if it matches the original expression.
[tex]\[
(2xy + 7)^2 = (2xy + 7)(2xy + 7)
\][/tex]
By expanding:
[tex]\[
= 2xy \cdot 2xy + 2xy \cdot 7 + 7 \cdot 2xy + 7 \cdot 7
\][/tex]
Simplify each term:
[tex]\[
= 4x^2y^2 + 14xy + 14xy + 49
\][/tex]
Combine like terms:
[tex]\[
= 4x^2y^2 + 28xy + 49
\][/tex]
5. Conclusion:
We see that the expanded form matches exactly with the given polynomial. Therefore:
[tex]\[
4x^2y^2 + 28xy + 49 = (2xy + 7)^2
\][/tex]
Thus, the factored form of the expression [tex]\(4x^2y^2 + 28xy + 49\)[/tex] is:
[tex]\[
(2xy + 7)^2
\][/tex]
