7. \( x^{2}+4 x+14=46 \)

To solve the equation \( x^{2}+4x+14=46 \), start by simplifying it. Subtract 46 from both sides: \[ x^{2}+4x+14-46=0 \] This simplifies to: \[ x^{2}+4x-32=0 \] Next, you can either factor the equation or use the quadratic formula. In this case, it can be factored as: \[ (x+8)(x-4)=0 \] This gives you two solutions: \[ x+8=0 \quad \Rightarrow \quad x=-8 \] \[ x-4=0 \quad \Rightarrow \quad x=4 \] So, the solutions are \( x = -8 \) and \( x = 4 \). To check your work, substituting both values back into the original equation should yield 46. In the world of algebra, it can be fascinating to see how quadratic equations represent parabolas on a graph. The roots, or solutions, reflect where the parabola intersects the x-axis. In this case, the parabola intersects at \( x = -8 \) and \( x = 4 \), creating a visual representation of these solutions! Don't forget the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), if you ever find yourself bumbling with a harder equation! It works universally for all quadratics and is your best friend when factoring gets tricky.