\( 300 + 40x + 30x + 4x^2 = 504 \) → \( 4x^2 + 70x - 204 = 0 \) → divide by 2: \( 2x^2 + 35x - 102 = 0 \)

["Solving the Quadratic Equation: (300 + 40x + 30x + 4x^2 = 504)", "If you're tackling a quadratic equation like (300 + 40x + 30x + 4x^2 = 504), simplifying it efficiently is key to finding its roots quickly and accurately. Let’s break down the step-by-step process and arrive at the simplified form and solution.", "---", "### Step 1: Combine Like Terms", "Start by combining the (x)-terms on the left-hand side:", "[\n4x^2 + (40x + 30x) + 300 = 504\n]", "[\n4x^2 + 70x + 300 = 504\n]", "---", "### Step 2: Move All Terms to One Side", "Subtract 504 from both sides to set the equation to zero:", "[\n4x^2 + 70x + 300 - 504 = 0\n]", "[\n4x^2 + 70x - 204 = 0\n]", "---", "### Step 3: Simplify the Equation", "To simplify, divide every term by 2:", "[\n\frac{4x^2}{2} + \frac{70x}{2} - \frac{204}{2} = 0\n]", "[\n2x^2 + 35x - 102 = 0\n]", "Now you have the elegant quadratic equation ready for solving.", "---", "### Solving (2x^2 + 35x - 102 = 0)", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 2), (b = 35), (c = -102):", "[\nx = \frac{-35 \pm \sqrt{35^2 - 4(2)(-102)}}{2(2)}\n]", "[\nx = \frac{-35 \pm \sqrt{1225 + 816}}{4}\n]", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "Since (\sqrt{2041}) is irrational, the roots remain in simplified radical form.", "---", "### Final Answer", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "---", "Summary\nSimplifying (300 + 40x + 30x + 4x^2 = 504) step-by-step leads to (2x^2 + 35x - 102 = 0), which can be solved using the quadratic formula. This process highlights the importance of combining like terms and simplifying coefficients to ease solving complex quadratics.", "---", "Keywords:\nquadratic equation, solve (2x^2 + 35x - 102 = 0), simplify (4x^2 + 70x - 204 = 0), solve (300 + 40x + 30x + 4x^2 = 504), quadratic formula, (x = \frac{-35 \pm \sqrt{2041}}{4}), algebra homework help."]