Solve: a²+7a+12=(a+?)(4+a) - Finding the Missing Factor

Fill in the missing number

$a^2+7a+12=(a+?)(4+a)$

To tackle this problem, we'll expand $(a + b)(4 + a)$ using the distributive property, compare it with the given quadratic equation $a^2 + 7a + 12$, and solve for the missing value $b$.

Step 1: Expand the expression $(a + b)(4 + a)$.

Applying the distributive property, we obtain:

$(a + b)(4 + a) = a(4 + a) + b(4 + a) = 4a + a^2 + 4b + ab$.

This simplifies to:

$a^2 + (4 + b)a + 4b$.

Step 2: Compare the expanded expression with $a^2 + 7a + 12$.

From the equation $a^2 + (4 + b)a + 4b = a^2 + 7a + 12$, equate the coefficients and constant term:

• For $a$: $4 + b = 7$
• For constant term: $4b = 12$

Step 3: Solve the equations.

• Solving $4 + b = 7$ yields $b = 3$.
• Additionally, $4b = 12$ also yields $b = 3$, confirming consistency.

Since both the conditions lead to $b = 3$, we verify the calculations.

Therefore, the missing number is $3$.