Factor each trinomial, if possible. See Examples 3 and 4. $32a^2+48ab+18b^2$

Verified step by step guidance

1. Identify the trinomial to factor: \$32a^2 + 48ab + 18b^2$.
2. Look for the greatest common factor (GCF) of all terms. The GCF of 32, 48, and 18 is 2, so factor out 2: \$2(16a^2 + 24ab + 9b^2)$.
3. Focus on factoring the quadratic trinomial inside the parentheses: \$16a^2 + 24ab + 9b^2$.
4. Use the method of factoring trinomials where the first term is a perfect square (\$16a^2 = (4a)^2\() and the last term is a perfect square (\)9b^2 = (3b)^2\(). Check if the middle term \)24ab\( equals \)2 imes 4a imes 3b$.
5. Since \$24ab = 2 imes 4a imes 3b\(, the trinomial inside the parentheses factors as a perfect square: \)(4a + 3b)^2\(. Therefore, the full factorization is \)2(4a + 3b)^2$.

Key Concepts

Factoring Trinomials

Factoring trinomials involves expressing a quadratic expression with three terms as a product of two binomials. This process often requires finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides all terms of a polynomial. Factoring out the GCF simplifies the trinomial, making it easier to factor further or identify special products.

Multiplying and Adding to Factor

When factoring trinomials with a leading coefficient other than 1, you multiply the first and last coefficients, then find two numbers that multiply to this product and add to the middle coefficient. These numbers help split the middle term for factoring by grouping.