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Factor. $\newline$ $3n^2 + 10n + 3$

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Factor quadratics with other leading coefficients

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Q. Factor.

\newline

3n^2 + 10n + 3

Identify $a$ , $b$ , $c$ : Identify $a$ , $b$ , and $c$ in the quadratic expression $3n^2 + 10n + 3$ . Compare $3n^2 + 10n + 3$ with the standard form $ax^2 + bx + c$ . $a = 3$ $b$ $0$ $b$ $1$
Find numbers for $a*c$ : Find two numbers that multiply to $a*c$ (which is $3*3=9$ ) and add up to $b$ (which is $10$ ). $\newline$ We need to find two numbers that satisfy these conditions. $\newline$ After checking possible factors of $9$ , we find that $1$ and $9$ are the numbers we are looking for because $1*9 = 9$ and $1+9 = 10$ .
Rewrite middle term: Rewrite the middle term $10n$ using the two numbers $1$ and $9$ found in Step $2$ . $\newline$ $3n^2 + 10n + 3$ can be rewritten as $3n^2 + n + 9n + 3$ .
Factor by grouping: Factor by grouping. $\newline$ Group the terms to factor out common factors: $\newline$ Group $3n^2 + n$ and $9n + 3$ . $\newline$ $3n^2 + n$ can be factored as $n(3n + 1)$ . $\newline$ $9n + 3$ can be factored as $3(3n + 1)$ .
Factor out common factor: Factor out the common binomial factor $(3n + 1)$ . $\newline$ We now have $n(3n + 1) + 3(3n + 1)$ . $\newline$ Factor out $(3n + 1)$ to get $(3n + 1)(n + 3)$ .

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