Check whether the following is a quadratic equation:
$x(2x + 3) = x^2 + 1$

(YES) Given equation: $x(2x + 3) = x^2 + 1$
First,expand the left-hand side $(LHS)$:
$x(2x + 3) = 2x^2 + 3x$
Now,substitute this back into the equation:
$2x^2 + 3x = x^2 + 1$
Subtract $(x^2 + 1)$ from both sides to bring the equation to the standard form $ax^2 + bx + c = 0$:
$2x^2 - x^2 + 3x - 1 = 0$
$x^2 + 3x - 1 = 0$
Since this equation is in the form $ax^2 + bx + c = 0$ where $a = 1$,$b = 3$,and $c = -1$ (and $a \neq 0$),the given equation is a quadratic equation.