Solve (x+?)(x-?) = x²-3x-40: Finding Missing Terms in Quadratic Factors

To solve this problem, we will follow these steps:

• Expand $(x+a)(x+b)$. This gives: $x^2 + (a+b)x + ab$.
• Compare with the quadratic equation $x^2 - 3x - 40$.
• Equate coefficients:

Now, work through each step:
1. The expanded form of $(x+a)(x+b)$ gives $x^2 + (a+b)x + ab$.
2. Comparing with $x^2 - 3x - 40$, we get two equations:
$a + b = -3$ (coefficient of $x$) and $ab = -40$ (constant term).

3. We need two numbers whose sum is $-3$ and product is $-40$.
4. Upon inspection, the numbers that satisfy these conditions are $a = 5$ and $b = -8$, since $5 + (-8) = -3$ and $5 \times (-8) = -40$.

Therefore, substituting $a$ and $b$ into the expression, the factorization of the quadratic is $(x+5)(x-8)$.

Thus, the solution to the problem is $\left( x + 5 \right) \left( x - 8 \right)$.