Sure, let's factorize the given expression step-by-step.
The given expression is:
[tex]\[ x^2 + 6xy + 9y^2 \][/tex]
Notice that the expression is a quadratic trinomial. To factorize it, we need to look for two binomials (in the form of [tex]\((ax + by)(cx + dy)\)[/tex]) whose product gives us the original quadratic trinomial.
1. Identify the first and the last term:
- The first term is [tex]\(x^2\)[/tex], which is [tex]\((x)(x)\)[/tex].
- The last term is [tex]\(9y^2\)[/tex], which is [tex]\((3y)(3y)\)[/tex].
2. Identify the middle term:
- The middle term is [tex]\(6xy\)[/tex].
3. Validate our factorization by checking the middle term:
- When we look at the possible binomials, we notice that if we use [tex]\((x + 3y)(x + 3y)\)[/tex], we need to check if this product gives us the correct middle term.
Now let's expand [tex]\((x + 3y)^2\)[/tex] to check if it matches the original expression:
[tex]\[
(x + 3y)(x + 3y) = x(x + 3y) + 3y(x + 3y)
\][/tex]
4. Distribute [tex]\(x\)[/tex] and [tex]\(3y\)[/tex]:
- [tex]\(x(x + 3y) = x^2 + 3xy\)[/tex]
- [tex]\(3y(x + 3y) = 3xy + 9y^2\)[/tex]
5. Combine like terms:
- [tex]\(x^2 + 3xy + 3xy + 9y^2 = x^2 + 6xy + 9y^2\)[/tex]
We can see that the expansion of [tex]\((x + 3y)^2\)[/tex] gives us back the original expression [tex]\(x^2 + 6xy + 9y^2\)[/tex].
Hence, the factorized form of the given expression is:
[tex]\[
(x + 3y)^2
\][/tex]
