The problem presents a quadratic equation $x^2 - 2(3k+1) + 7(2k+3) = 0$. We need to rewrite the equation and solve for $x$. However, the equation provided has an error. It appears there is an 'x' missing from the term $2(3k+1)x$. Assuming the correct equation is $x^2 - 2(3k+1)x + 7(2k+3) = 0$, we can proceed to solve this equation for $x$ using the quadratic formula.

1. Problem Description

The problem presents a quadratic equation $x^2 - 2(3k+1) + 7(2k+3) = 0$. We need to rewrite the equation and solve for $x$. However, the equation provided has an error. It appears there is an 'x' missing from the term $2(3k+1)x$.

Assuming the correct equation is $x^2 - 2(3k+1)x + 7(2k+3) = 0$, we can proceed to solve this equation for $x$ using the quadratic formula.

2. Solution Steps

The given quadratic equation is $x^2 - 2(3k+1)x + 7(2k+3) = 0$.

We can rewrite this equation as $x^2 - (6k+2)x + (14k+21) = 0$.

Now, we use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = -(6k+2)$, and $c = 14k+21$.

Substitute these values into the quadratic formula:

$x = \frac{6k+2 \pm \sqrt{(-(6k+2))^2 - 4(1)(14k+21)}}{2(1)}$

$x = \frac{6k+2 \pm \sqrt{(6k+2)^2 - 4(14k+21)}}{2}$

$x = \frac{6k+2 \pm \sqrt{36k^2 + 24k + 4 - 56k - 84}}{2}$

$x = \frac{6k+2 \pm \sqrt{36k^2 - 32k - 80}}{2}$

$x = \frac{6k+2 \pm \sqrt{4(9k^2 - 8k - 20)}}{2}$

$x = \frac{6k+2 \pm 2\sqrt{9k^2 - 8k - 20}}{2}$

$x = 3k+1 \pm \sqrt{9k^2 - 8k - 20}$

3. Final Answer

$x = 3k+1 \pm \sqrt{9k^2 - 8k - 20}$