We are asked to solve the following equation for $x$: $$(2x+1)^2 - 4(x-3)^2 = 5x+10$$

1. Problem Description

We are asked to solve the following equation for $x$:

$$(2x+1)^2 - 4(x-3)^2 = 5x+10$$

2. Solution Steps

First, expand the squared terms:

$(2x+1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1$

$(x-3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$

Substitute these expressions back into the original equation:

$$(4x^2 + 4x + 1) - 4(x^2 - 6x + 9) = 5x + 10$$

$$4x^2 + 4x + 1 - 4x^2 + 24x - 36 = 5x + 10$$

Simplify the equation:

$$(4x^2 - 4x^2) + (4x + 24x) + (1 - 36) = 5x + 10$$

$$28x - 35 = 5x + 10$$

Subtract $5x$ from both sides:

$$28x - 5x - 35 = 5x - 5x + 10$$

$$23x - 35 = 10$$

Add 35 to both sides:

$$23x - 35 + 35 = 10 + 35$$

$$23x = 45$$

Divide both sides by 23:

$$x = \frac{45}{23}$$

3. Final Answer

The final answer is $x = \frac{45}{23}$.