Question

factor.
w^{2}+19w + 48

Explanation:

Step1: Find two numbers that multiply to 48 and add to 19.

We need two numbers \(a\) and \(b\) such that \(a\times b = 48\) and \(a + b=19\). The factors of 48 are: 1 and 48 (sum 49), 2 and 24 (sum 26), 3 and 16 (sum 19), 4 and 12 (sum 16), 6 and 8 (sum 14). So the numbers are 3 and 16.

Step2: Rewrite the middle term using these numbers.

\(w^{2}+19w + 48=w^{2}+3w+16w + 48\)

Step3: Group the terms and factor.

Group the first two and last two terms: \((w^{2}+3w)+(16w + 48)\). Factor out the GCF from each group: \(w(w + 3)+16(w + 3)\). Now factor out the common binomial factor \((w + 3)\): \((w + 3)(w + 16)\)

Answer:

\((w + 3)(w + 16)\)