Given: \( 36=x+4 x^{2}+8 \), find \( a, b \), and \( c \). \( \begin{array}{l}\text { a. } a=1, b=4, c=-28 \\ \text { b. } a=1, b=4, c=8 \\ \text { c. } a=-3, b=1, c=8 \\ \text { d. } a=4, b=1, c=-28\end{array} \)

To solve the equation \( 36 = x + 4x^2 + 8 \), let's first rearrange it into standard quadratic form \( ax^2 + bx + c = 0 \). Starting with the equation: \[ 4x^2 + x + 8 - 36 = 0 \] This simplifies to: \[ 4x^2 + x - 28 = 0 \] Now we can identify the coefficients: - \( a = 4 \) - \( b = 1 \) - \( c = -28 \) So, the correct option is: **d. \( a=4, b=1, c=-28 \)** --- Here's an additional fun fact: Quadratic equations can often be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula not only gives you the solutions but also the insight into the relationship between the coefficients and the roots of the equation. Another amusing point: Quadratics are like the bread and butter of math; they've been around for over two millennia! Ancient Babylonian mathematicians were already using algebraic methods that resemble our modern understanding of solving squares. So, every time you solve a quadratic, you’re tapping into thousands of years of mathematical history!