Question

factor.
$y^2 + 10y + 21$

Explanation:

Step1: Find two numbers that multiply to 21 and add to 10.

We need two numbers \(a\) and \(b\) such that \(a\times b = 21\) and \(a + b=10\). The factors of 21 are 1 & 21, 3 & 7. Among these, \(3 + 7=10\) and \(3\times7 = 21\).

Step2: Rewrite the middle term using these numbers.

We can rewrite the quadratic expression \(y^{2}+10y + 21\) as \(y^{2}+3y+7y + 21\).

Step3: Group the terms and factor by grouping.

Group the first two terms and the last two terms: \((y^{2}+3y)+(7y + 21)\).
Factor out the greatest common factor from each group: \(y(y + 3)+7(y + 3)\).

Step4: Factor out the common binomial factor.

Now, we can factor out \((y + 3)\) from both terms: \((y + 3)(y + 7)\).

Answer:

\((y + 3)(y + 7)\)