Factor the expression by grouping. First, the expression needs to be rewritten as $3x^{2}+ax+bx+4$. To find $a$ and $b$, set up a system to be solved.
$$a+b=7$$ $$ab=3\times 4=12$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. List all such integer pairs that give product $12$.
$$1,12$$ $$2,6$$ $$3,4$$
Calculate the sum for each pair.
$$1+12=13$$ $$2+6=8$$ $$3+4=7$$
The solution is the pair that gives sum $7$.
$$a=3$$ $$b=4$$
Rewrite $3x^{2}+7x+4$ as $\left(3x^{2}+3x\right)+\left(4x+4\right)$.
$$\left(3x^{2}+3x\right)+\left(4x+4\right)$$
Factor out $3x$ in the first and $4$ in the second group.
$$3x\left(x+1\right)+4\left(x+1\right)$$
Factor out common term $x+1$ by using distributive property.
$$\left(x+1\right)\left(3x+4\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$3x^{2}+7x+4=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $x=\frac{-7±1}{6}$ when $±$ is plus. Add $-7$ to $1$.
$$x=-\frac{6}{6}$$
Divide $-6$ by $6$.
$$x=-1$$
Now solve the equation $x=\frac{-7±1}{6}$ when $±$ is minus. Subtract $1$ from $-7$.
$$x=-\frac{8}{6}$$
Reduce the fraction $\frac{-8}{6}$ to lowest terms by extracting and canceling out $2$.
$$x=-\frac{4}{3}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-1$ for $x_{1}$ and $-\frac{4}{3}$ for $x_{2}$.
Cancel out $3$, the greatest common factor in $3$ and $3$.
$$3x^{2}+7x+4=\left(x+1\right)\left(3x+4\right)$$
Steps Using Direct Factoring Method
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form $x^2+Bx+C=0$.This is achieved by dividing both sides of the equation by $3$
$$x ^ 2 +\frac{7}{3}x +\frac{4}{3} = 0$$
Let $r$ and $s$ be the factors for the quadratic equation such that $x^2+Bx+C=(x−r)(x−s)$ where sum of factors $(r+s)=−B$ and the product of factors $rs = C$
$$r + s = -\frac{7}{3} $$ $$ rs = \frac{4}{3}$$
Two numbers $r$ and $s$ sum up to $-\frac{7}{3}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{7}{3} = -\frac{7}{6}$. You can also see that the midpoint of $r$ and $s$ corresponds to the axis of symmetry of the parabola represented by the quadratic equation $y=x^2+Bx+C$. The values of $r$ and $s$ are equidistant from the center by an unknown quantity $u$. Express $r$ and $s$ with respect to variable $u$.
$$r = -\frac{7}{6} - u$$ $$s = -\frac{7}{6} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = \frac{4}{3}$
