Sure! Let's fully factorize the expression [tex]\( x^2 + 8xy + 12y^2 \)[/tex] step by step.
1. Identify the form of the quadratic expression:
The given expression is a quadratic trinomial in the form [tex]\( ax^2 + bxy + cy^2 \)[/tex] with:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = 12 \)[/tex]
2. Look for two binomials of the form [tex]\((x + m y)(x + n y)\)[/tex]:
We need to find integers [tex]\(m\)[/tex] and [tex]\(n\)[/tex] such that:
- [tex]\( m \cdot n = c = 12 \)[/tex]
- [tex]\( m + n = b = 8 \)[/tex]
3. Find pairs of factors of [tex]\(c = 12\)[/tex] that add up to [tex]\(b = 8\)[/tex] and test them:
Let's determine the pairs that multiply to 12:
- (1, 12)
- (2, 6)
- (3, 4)
- (-1, -12)
- (-2, -6)
- (-3, -4)
Out of these pairs, the pair (2, 6) sums up to 8 (i.e., [tex]\( 2 + 6 = 8 \)[/tex]).
4. Write the expression in factored form:
Given the pairs (2, 6), we can rewrite the expression [tex]\( x^2 + 8xy + 12y^2 \)[/tex] as:
[tex]\[
(x + 2y)(x + 6y)
\][/tex]
5. Verification:
To verify whether our factorization is correct, we can expand [tex]\((x + 2y)(x + 6y)\)[/tex] and check if we get back the original expression:
[tex]\[
(x + 2y)(x + 6y) = x \cdot x + x \cdot 6y + 2y \cdot x + 2y \cdot 6y = x^2 + 6xy + 2xy + 12y^2 = x^2 + 8xy + 12y^2
\][/tex]
Since we get back the original expression, our factorization is correct.
Therefore, the fully factorized form of [tex]\( x^2 + 8xy + 12y^2 \)[/tex] is:
[tex]\[
(x + 2y)(x + 6y)
\][/tex]
