Question

factor completely. $3x^2 + 10x + 7$

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(3\times7 = 21\) and add up to \(10\). The numbers are \(3\) and \(7\) since \(3\times7 = 21\) and \(3 + 7=10\).

Step2: Split the middle term

Rewrite the middle term \(10x\) as \(3x+7x\):
\(3x^{2}+3x + 7x+7\)

Step3: Group and factor

Group the first two terms and the last two terms:
\((3x^{2}+3x)+(7x + 7)\)
Factor out the common factors from each group:
\(3x(x + 1)+7(x + 1)\)

Step4: Factor out the common binomial

Now, factor out the common binomial \((x + 1)\):
\((3x + 7)(x + 1)\)

Answer:

\((3x + 7)(x + 1)\)