Solve (x+?)(x+?) = x²+7x+10: Finding Missing Terms in Quadratic Factors

To solve this problem, we'll follow these steps:

• Step 1: Identify the pattern of factoring quadratics.
• Step 2: Use the sum and product relation to find the numbers.
• Step 3: Confirm the binomial expressions.

Let's work through each step:

Step 1: Our task is to factor the expression $x^2 + 7x + 10$. We need two numbers whose product is $10$ (the constant term) and whose sum is $7$ (the coefficient of $x$).

Step 2: We list pairs of numbers that multiply to $10$:
- $1 \times 10$
- $2 \times 5$

Among these, only the pair $2$ and $5$ add up to $7$. Therefore, the expressions needed are $(x+2)$ and $(x+5)$.

Step 3: We substitute these values into the binomials: $(x+5)(x+2)$. Expanding this verifies:
$(x+5)(x+2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10$.

Therefore, the factorization is correct, and the solution reached is $\left(x+5\right)\left(x+2\right)$.

Therefore, the solution to the problem is $\left(x+5\right)\left(x+2\right)$.