\( h(2) = 15 \) gives us \( a(2)^2 + b(2) + 3 = 15 \) which simplifies to \( 4a + 2b + 3 = 15 \). Thus, \( 4a + 2b = 12 \).

["Understanding Quadratic Equations: How ( h(2) = 15 ) Shapes Our Equation", "When working with quadratic functions, a common scenario arises where you evaluate a function at a specific input value—say, ( h(2) = 15 )—and end up confronting a simple but powerful algebraic relationship. Take the equation ( a(2)^2 + b(2) + 3 = 15 ). At first glance, this may look like just a numerical substitution, but behind it lies a gateway to understanding coefficients, function behavior, and real-world modeling.", "### From Function Evaluation to Simplified Form", "Let’s start with the underlying equation:", "[\nh(2) = a(2)^2 + b(2) + 3 = 15\n]", "This reflects the output of the quadratic function when the input is ( x = 2 ). Substituting ( x = 2 ), we get:", "[\na(2) = a \cdot 2^2 = 4a, \quad b(2) = b \cdot 2 = 2b\n]", "So the original equation becomes:", "[\n4a + 2b + 3 = 15\n]", "Subtracting 3 from both sides simplifies it to:", "[\n4a + 2b = 12\n]", "This linear equation in two variables is often the first step toward identifying relationships between ( a ) and ( b ). But why does it matter?", "### Why This Equation Matters in Algebra and Applications", "This simplified form ( 4a + 2b = 12 ) is more than just arithmetic—it reveals proportionality and constraints of the system. Dividing every term by 2 gives:", "[\n2a + b = 6\n]", "This equation shows how coefficients ( a ) and ( b ) are interrelated for ( h(2) = 15 ). Whether you’re fitting a curve to data, modeling population growth, or analyzing physical systems like projectile motion, such relationships allow us to solve for unknowns efficiently.", "### Practical Use in Problem Solving", "For example, suppose you’re tasked with finding integer values of ( a ) and ( b ) satisfying this condition. Rewriting ( b = 6 - 2a ), you see that ( a ) determines ( b )—a foundation for generation of valid pairs. This principle applies widely: in machine learning ( \vec{a} ) and ( \vec{b} ) parameters constrained by loss functions, or in optimization where function outputs determine feasible regions.", "### Conclusion", "The journey from ( h(2) = 15 ) into the equation ( 4a + 2b = 12 ) illustrates a fundamental algebraic process—transforming a subfunction evaluation into a solvable linear constraint. Understanding this connection empowers students and professionals alike to decode function behavior, solve systems efficiently, and apply mathematical reasoning across science and engineering disciplines.", "So next time you see ( 4a + 2b = 12 ), remember: it’s not just coefficients on a page—it’s a clue to the dynamics of quadratic relationships.", "---", "Keywords: ( h(2) = 15 ), quadratic function evaluation, ( a(2)^2 + b(2) + 3 = 15 ), function parameterization, solving for ( a ) and ( b ), ( 4a + 2b = 12 ), algebra simplification, quadratic modeling, mathematical relationships."]