Answer
$$5(x + 7)(x - 3)$$
Work Step by Step
To factor this polynomial, we need to factor out the greatest common factor of both the coefficients and the constant:
The greatest common factor of the coefficients and constant $5$, $20$, and $105$ is $5$.
Therefore, we will factor out $5$ from each term to get:
$$5(x^2 + 4x - 21)$$
We see that we now have a polynomial that we can factor even further. We want to find out which combination of factors for $-21$ will add up to $4$. Let us look at the factors for $-21$:
$(-21)$ and $(1)$ or $(21)$ and $(-1)$
$(-7)$ and $(3)$ or $(7)$ and $(-3)$
We want a bigger positive number and a smaller negative factor so that their addition will give a positive number, so we should choose $7$ and $-3$:
$$5(x + 7)(x - 3)$$
If we want to check if we had factored correctly, we multiply the binomials using the FOIL method to see if we get the same original equation:
$$5[(x)(x) - 3(x) + 7(x) + (7)(-3)]$$
Multiplying the terms, we get:
$$5[(x)(x) - 3(x) + 7(x) + (7)(-3)]$$
$$5(x^2 - 3x + 7x - 21)$$
Combine like terms:
$$5(x^2 + 4x - 21)$$
Multiply $5$ with each of the terms of the trinomial:
$$5x^2 + 20x - 105$$
We see that we get the same original equation, so we know that our factoring was correct.
