Factor the expression completely, or state that the polynomial is prime: 7x² − 7x − 140.

Step-by-step solution

1. 1

   Look for a common factor before anything else. All three coefficients $7$, $-7$ and $-140$ are divisible by $7$, so factoring it out first keeps the numbers small: $7x^2 - 7x - 140 = 7(x^2 - x - 20)$.

2. 2

   Set up the trinomial search. Since $x^2 - x - 20$ is monic, we need two integers whose product is $-20$ and whose sum is $-1$, the coefficient of $x$.

3. 3

   Find the pair. The factor pairs of $-20$ are $(1,-20)$, $(2,-10)$, $(4,-5)$ and their sign swaps. Only $-5$ and $4$ add to $-1$.

4. 4

   Write the binomials. That gives $x^2 - x - 20 = (x - 5)(x + 4)$, so the full factorisation is $7(x - 5)(x + 4)$.

5. 5

   Confirm by expanding. $(x-5)(x+4) = x^2 - x - 20$, and multiplying by $7$ returns $7x^2 - 7x - 140$, the original expression.