Question
Factor completely.
2m2+9m+7
2m2+9m+7
Answers
GPT-4o mini
To factor the quadratic expression \(2m^2 + 9m + 7\) completely, we will look for two numbers that multiply to \(2 \times 7 = 14\) (the product of the leading coefficient and the constant term) and add to \(9\) (the middle coefficient).
The numbers that fit these criteria are \(7\) and \(2\), since:
- \(7 \times 2 = 14\)
- \(7 + 2 = 9\)
Now we'll rewrite the middle term \(9m\) using \(7m\) and \(2m\):
\[
2m^2 + 7m + 2m + 7
\]
Next, we can group the terms:
\[
(2m^2 + 7m) + (2m + 7)
\]
Now we factor out the common factors in each group:
\[
m(2m + 7) + 1(2m + 7)
\]
We can see that \(2m + 7\) is a common factor:
\[
(2m + 7)(m + 1)
\]
Thus, the expression \(2m^2 + 9m + 7\) factors completely to:
\[
\boxed{(2m + 7)(m + 1)}
\]
The numbers that fit these criteria are \(7\) and \(2\), since:
- \(7 \times 2 = 14\)
- \(7 + 2 = 9\)
Now we'll rewrite the middle term \(9m\) using \(7m\) and \(2m\):
\[
2m^2 + 7m + 2m + 7
\]
Next, we can group the terms:
\[
(2m^2 + 7m) + (2m + 7)
\]
Now we factor out the common factors in each group:
\[
m(2m + 7) + 1(2m + 7)
\]
We can see that \(2m + 7\) is a common factor:
\[
(2m + 7)(m + 1)
\]
Thus, the expression \(2m^2 + 9m + 7\) factors completely to:
\[
\boxed{(2m + 7)(m + 1)}
\]