The problem is to solve the quadratic equation $7x^2 + 4 = 11x$ by factoring.

1. Problem Description

The problem is to solve the quadratic equation $7x^2 + 4 = 11x$ by factoring.

2. Solution Steps

First, we rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$:

$7x^2 - 11x + 4 = 0$.

Now, we need to factor the quadratic expression $7x^2 - 11x + 4$. We are looking for two numbers that multiply to $7 \cdot 4 = 28$ and add up to $-11$. These two numbers are $-4$ and $-7$. We can rewrite the middle term as $-4x - 7x$:

$7x^2 - 7x - 4x + 4 = 0$.

Now, we factor by grouping:

$7x(x - 1) - 4(x - 1) = 0$.

We can factor out the common term $(x - 1)$:

$(7x - 4)(x - 1) = 0$.

Now, we set each factor equal to zero and solve for $x$:

$7x - 4 = 0 \implies 7x = 4 \implies x = \frac{4}{7}$

$x - 1 = 0 \implies x = 1$

So, the solutions are $x = \frac{4}{7}$ and $x = 1$.

3. Final Answer

$x = \frac{4}{7}, 1$