1. The first problem given is to simplify or analyze the expression $a^2 + 53a - 14$. 2. This is a quadratic expression in standard form $ax^2 + bx + c$ where $a=1$, $b=53$, and $c=-14$. 3. To factor or solve this quadratic, we can use the quadratic formula: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. Substitute the values: $$a = \frac{-53 \pm \sqrt{53^2 - 4 \times 1 \times (-14)}}{2 \times 1}$$ 5. Calculate the discriminant: $$53^2 = 2809$$ $$4 \times 1 \times (-14) = -56$$ $$\text{Discriminant} = 2809 - (-56) = 2809 + 56 = 2865$$ 6. Find the square root of the discriminant: $$\sqrt{2865} \approx 53.53$$ 7. Calculate the two roots: $$a_1 = \frac{-53 + 53.53}{2} = \frac{0.53}{2} = 0.265$$ $$a_2 = \frac{-53 - 53.53}{2} = \frac{-106.53}{2} = -53.265$$ 8. Therefore, the solutions to the quadratic equation $a^2 + 53a - 14 = 0$ are approximately: $$a \approx 0.265 \text{ or } a \approx -53.265$$ 9. If the goal is to factor the expression, since the discriminant is not a perfect square, the quadratic does not factor nicely over the integers. 10. Summary: The quadratic $a^2 + 53a - 14$ has roots approximately $0.265$ and $-53.265$ found using the quadratic formula.