To factor the polynomial [tex]\(4x^2 + y^2 + 4xy + 8x + 4y + 4\)[/tex], let us follow the steps below:
1. Rewrite the polynomial to identify common terms and structure:
[tex]\[
4x^2 + y^2 + 4xy + 8x + 4y + 4
\][/tex]
2. Group the terms to recognize a possible complete square:
Notice that the polynomial contains both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms as well as mixed terms like [tex]\(4xy\)[/tex]. We can guess it might be a square of a binomial due to the structure.
3. Consider the expressions
[tex]\[
(2x + y + a)^2
\][/tex]
This expansion would look something like:
[tex]\[
(2x + y + a)^2 = (2x)^2 + (y)^2 + 2(2x)(y) + 2(2x)(a) + 2(y)(a) + a^2
\][/tex]
Expanding it out:
[tex]\[
4x^2 + y^2 + 4xy + 4ax + 2ay + a^2
\][/tex]
4. Compare and solve for 'a':
To match our original polynomial [tex]\(4x^2 + y^2 + 4xy + 8x + 4y + 4\)[/tex], the coefficients of the terms should match:
[tex]\[
4ax = 8x \quad \text{which gives } a = 2
\][/tex]
[tex]\[
2ay = 4y \quad \text{which also confirms } a = 2
\][/tex]
[tex]\[
a^2 = 4 \quad \text{(which indeed is } 2^2)
\][/tex]
5. Form the complete square:
Thus, [tex]\(a\)[/tex] matches our required condition. Placing [tex]\(a = 2\)[/tex] into the expression we get:
[tex]\[
(2x + y + 2)^2
\][/tex]
6. Factorized form of the polynomial:
Therefore, the factorization of the polynomial [tex]\(4x^2 + y^2 + 4xy + 8x + 4y + 4\)[/tex] is:
[tex]\[
(2x + y + 2)^2
\][/tex]
So, the factor of the polynomial [tex]\(4x^2 + y^2 + 4xy + 8x + 4y + 4\)[/tex] is [tex]\((2x + y + 2)^2\)[/tex].
