Factor the expression by grouping. First, the expression needs to be rewritten as $12k^{2}+ak+bk+15$. To find $a$ and $b$, set up a system to be solved.
$$a+b=28$$ $$ab=12\times 15=180$$
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 $180$.
Rewrite $12k^{2}+28k+15$ as $\left(12k^{2}+10k\right)+\left(18k+15\right)$.
$$\left(12k^{2}+10k\right)+\left(18k+15\right)$$
Factor out $2k$ in the first and $3$ in the second group.
$$2k\left(6k+5\right)+3\left(6k+5\right)$$
Factor out common term $6k+5$ by using distributive property.
$$\left(6k+5\right)\left(2k+3\right)$$
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 $ax^{2}+bx+c=0$.
$$12k^{2}+28k+15=0$$
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 $k=\frac{-28±8}{24}$ when $±$ is plus. Add $-28$ to $8$.
$$k=-\frac{20}{24}$$
Reduce the fraction $\frac{-20}{24}$ to lowest terms by extracting and canceling out $4$.
$$k=-\frac{5}{6}$$
Now solve the equation $k=\frac{-28±8}{24}$ when $±$ is minus. Subtract $8$ from $-28$.
$$k=-\frac{36}{24}$$
Reduce the fraction $\frac{-36}{24}$ to lowest terms by extracting and canceling out $12$.
$$k=-\frac{3}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-\frac{5}{6}$ for $x_{1}$ and $-\frac{3}{2}$ for $x_{2}$.
Multiply $\frac{6k+5}{6}$ times $\frac{2k+3}{2}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
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 $12$
$$x ^ 2 +\frac{7}{3}x +\frac{5}{4} = 0$$
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 $rs = C$
$$r + s = -\frac{7}{3} $$ $$ rs = \frac{5}{4}$$
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 $rs = \frac{5}{4}$
