1. We are given the equation $(n + 90)(n - 10) = 475$ and asked to solve for $n$. 2. First, expand the left side using the distributive property: $$ (n + 90)(n - 10) = n^2 - 10n + 90n - 900 = n^2 + 80n - 900 $$ 3. Set the equation equal to 475: $$ n^2 + 80n - 900 = 475 $$ 4. Subtract 475 from both sides to set the equation to zero: $$ n^2 + 80n - 900 - 475 = 0 $$ $$ n^2 + 80n - 1375 = 0 $$ 5. Use the quadratic formula to solve $n^2 + 80n - 1375 = 0$, where $a=1$, $b=80$, and $c=-1375$: $$ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-80 \pm \sqrt{80^2 - 4 \times 1 \times (-1375)}}{2 \times 1} $$ 6. Calculate the discriminant: $$ 80^2 - 4 \times 1 \times (-1375) = 6400 + 5500 = 11900 $$ 7. Simplify the square root: $$ \sqrt{11900} = \sqrt{100 \times 119} = 10\sqrt{119} $$ 8. Substitute back: $$ n = \frac{-80 \pm 10\sqrt{119}}{2} = -40 \pm 5\sqrt{119} $$ 9. Therefore, the two solutions are: $$ n = -40 + 5\sqrt{119} $$ $$ n = -40 - 5\sqrt{119} $$ These are the exact forms of the solutions to the equation.