Find Missing Factors in (4x+8)(? + ?) = 4ax+8a+12x+24

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

• Step 1: Apply the distributive property to $(4x+8)(b+c)$
• Step 2: Match the expanded terms to $4ax + 8a + 12x + 24$
• Step 3: Solve for $b$ and $c$

Now, let's work through each step:
Step 1: Use the distributive property to expand $(4x+8)(b+c)$. This gives us:

$(4x+8)(b+c) = 4x \cdot b + 8 \cdot b + 4x \cdot c + 8 \cdot c$

Step 2: Equate the expression from Step 1 to $4ax + 8a + 12x + 24$:
$4bx + 8b + 4cx + 8c = 4ax + 8a + 12x + 24$

Separate and equate the coefficients for $x$ and the constant terms:

• For $x$: $4b + 4c = 4a + 12$
• For constants: $8b + 8c = 8a + 24$

Step 3: Solve the resulting system of equations:

Divide each equation by its common factor: - $4b + 4c = 4a + 12$ becomes: $b + c = a + 3$ - $8b + 8c = 8a + 24$ becomes: $b + c = a + 3$

Both equations are identical, thus we only need one further condition to solve completely.

Match assumptions based on simplest composition of terms:
Assume $b = a$ and $c = 3$ to verify this works correctly:

Substituting these into $b + c = a + 3$ gives:
$a + 3 = a + 3$, confirming our choice is consistent.

Thus, the solution to the problem for missing values is $a,3$.