1. The problem is to factor the quadratic expression $x^2 - 7x - 78$. 2. The general form of a quadratic expression is $ax^2 + bx + c$. Here, $a=1$, $b=-7$, and $c=-78$. 3. To factor, we look for two numbers that multiply to $ac = 1 \times (-78) = -78$ and add to $b = -7$. 4. The pair of numbers that satisfy this are $6$ and $-13$ because $6 \times (-13) = -78$ and $6 + (-13) = -7$. 5. Rewrite the middle term using these numbers: $$x^2 + 6x - 13x - 78$$ 6. Group terms: $$(x^2 + 6x) + (-13x - 78)$$ 7. Factor each group: $$x(x + 6) - 13(x + 6)$$ 8. Factor out the common binomial: $$(x - 13)(x + 6)$$ 9. Therefore, the factored form of $x^2 - 7x - 78$ is $$(x - 13)(x + 6)$$.