Problem: Factor $x^2 + 5x - 6$. 1. Identify the coefficients for the quadratic in the form $ax^2 + bx + c$: we have $a=1$, $b=5$, and $c=-6$. 2. Compute the product $ac = 1\cdot -6 = -6$ and note we need two numbers that multiply to $-6$ and add to $b=5$. 3. Find the pair $6$ and $-1$ since $6\cdot -1 = -6$ and $6 + (-1) = 5$. 4. Split the middle term using these numbers to rewrite the polynomial as $x^2 + 6x - x - 6$. 5. Factor by grouping: group into $ (x^2 + 6x) + (-x - 6)$. 6. Factor each group to get $x(x+6) -1(x+6)$. 7. Factor out the common factor $x+6$ to obtain $ (x+6)(x-1)$. 8. Check by expanding: $ (x+6)(x-1) = x^2 - x + 6x -6 = x^2 +5x -6$, which matches the original polynomial. Final answer: The factorization is $ (x+6)(x-1)$.