Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx-42$. To find $a$ and $b$, set up a system to be solved.
$$a+b=11$$ $$ab=1\left(-42\right)=-42$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-42$.
$$-1,42$$ $$-2,21$$ $$-3,14$$ $$-6,7$$
Calculate the sum for each pair.
$$-1+42=41$$ $$-2+21=19$$ $$-3+14=11$$ $$-6+7=1$$
The solution is the pair that gives sum $11$.
$$a=-3$$ $$b=14$$
Rewrite $x^{2}+11x-42$ as $\left(x^{2}-3x\right)+\left(14x-42\right)$.
$$\left(x^{2}-3x\right)+\left(14x-42\right)$$
Factor out $x$ in the first and $14$ in the second group.
$$x\left(x-3\right)+14\left(x-3\right)$$
Factor out common term $x-3$ by using distributive property.
$$\left(x-3\right)\left(x+14\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}+11x-42=0$$
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.
