Question: If $ a \cdot (a + b) = a^2 + 12 $, and $ a = 4 $, what is the value of $ b $?

["Question: If $ a \cdot (a + b) = a^2 + 12 $, and $ a = 4 $, what is the value of $ b $?", "When solving mathematical equations, substituting known values is a key step to uncovering unknowns. In this case, we are given the equation:\n$$ a \cdot (a + b) = a^2 + 12 $$\nwith the value $ a = 4 $. Substituting $ a = 4 $ into the equation allows us to isolate $ b $ and find its value efficiently.", "### Step 1: Substitute $ a = 4 $ into the equation", "Replace every instance of $ a $ with 4:\n$$ 4 \cdot (4 + b) = 4^2 + 12 $$", "### Step 2: Simplify both sides", "Calculate $ 4^2 $:\n$$ 4^2 = 16 $$", "So the equation becomes:\n$$ 4 \cdot (4 + b) = 16 + 12 $$\n$$ 4 \cdot (4 + b) = 28 $$", "### Step 3: Solve for $ b $", "Divide both sides by 4 to isolate the parentheses:\n$$ 4 + b = \frac{28}{4} $$\n$$ 4 + b = 7 $$", "Now subtract 4 from both sides:\n$$ b = 7 - 4 $$\n$$ b = 3 $$", "### Final Answer", "So, if $ a \cdot (a + b) = a^2 + 12 $ and $ a = 4 $, then the value of $ b $ is 3.", "This simple algebraic manipulation illustrates how substitution and basic arithmetic help solve equations—key skills for students and anyone working with equations daily. Understanding how to solve for a variable step-by-step builds a strong foundation in algebra and problem-solving.", "Keywords: algebra, solve for b, equation solving, $ a \cdot (a + b) = a^2 + 12 $, how to find b, step-by-step math, math tutorial, basic algebra."]