Given the matrices M = eginpmatrix 3 & -2 \ -4 & 3 endpmatrix and N = eginpmatrix 5 & 4 \ -2 & 6 endpmatrix, find a 2 x 2 matrix, P, such that PM = N.

We are given the equation $PM = N$. To find matrix $P$, we need to multiply both sides of the equation by the inverse of $M$, denoted as $M^{-1}$, on the right side.

$$PM M^{-1} = N M^{-1}$$

Since $M M^{-1} = I$ (the identity matrix) and $PI = P$, the equation simplifies to:

$$P = N M^{-1}$$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $ad - bc$.
For $M = \begin{pmatrix} 3 & -2 \\ -4 & 3 \end{pmatrix}$:

$$\det(M) = (3)(3) - (-2)(-4) = 9 - 8 = 1$$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the inverse is $\frac{1}{\det(M)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

$$M^{-1} = \frac{1}{1} \begin{pmatrix} 3 & -(-2) \\ -(-4) & 3 \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$$

$$P = N M^{-1} = \begin{pmatrix} 5 & 4 \\ -2 & 6 \end{pmatrix} \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$$

Perform the matrix multiplication:

$$P = \begin{pmatrix} (5)(3) + (4)(4) & (5)(2) + (4)(3) \\ (-2)(3) + (6)(4) & (-2)(2) + (6)(3) \end{pmatrix}$$ $$P = \begin{pmatrix} 15 + 16 & 10 + 12 \\ -6 + 24 & -4 + 18 \end{pmatrix}$$ $$P = \begin{pmatrix} 31 & 22 \\ 18 & 14 \end{pmatrix}$$

The matrix $P$ is $\boxed{\begin{pmatrix} 31 & 22 \\ 18 & 14 \end{pmatrix}}$.