The problem is to factor the quadratic expression $-4x^2 + 23x + 6$.

1. Problem Description

The problem is to factor the quadratic expression $-4x^2 + 23x + 6$.

2. Solution Steps

To factor the quadratic expression $-4x^2 + 23x + 6$, we can first factor out a $-1$ to get $-(4x^2 - 23x - 6)$.

Now, we need to factor the quadratic $4x^2 - 23x - 6$.

We are looking for two numbers that multiply to $4(-6) = -24$ and add up to $-23$. These numbers are $-24$ and $1$.

We can rewrite the middle term as:

$4x^2 - 23x - 6 = 4x^2 - 24x + x - 6$

Now, we factor by grouping:

$4x^2 - 24x + x - 6 = 4x(x - 6) + 1(x - 6)$

We can factor out $(x - 6)$:

$4x(x - 6) + 1(x - 6) = (4x + 1)(x - 6)$

Therefore, $4x^2 - 23x - 6 = (4x + 1)(x - 6)$.

Since we factored out a $-1$ in the beginning, we have:

$-4x^2 + 23x + 6 = -(4x^2 - 23x - 6) = -(4x + 1)(x - 6) = (-4x - 1)(x - 6)$

Alternatively, we can write it as $(4x + 1)(-x + 6)$.

3. Final Answer

The factored form of the given expression is $(4x+1)(-x+6)$.