Question

Question was provided via image upload.

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. So, we have the proportion \(\frac{4}{x - 4}=\frac{x + 3}{(x + 14)-(x + 3)}\).
Simplify the denominator on the right - hand side: \((x + 14)-(x + 3)=x+14 - x - 3 = 11\). The proportion becomes \(\frac{4}{x - 4}=\frac{x + 3}{11}\).

Step2: Cross - multiply

Cross - multiplying gives us \(4\times11=(x + 3)(x - 4)\).
Expand the right - hand side using the FOIL method: \((x + 3)(x - 4)=x^{2}-4x+3x - 12=x^{2}-x - 12\). So, \(44=x^{2}-x - 12\).

Step3: Rearrange into standard quadratic form

Rearrange the equation to \(x^{2}-x - 12-44 = 0\), which simplifies to \(x^{2}-x - 56 = 0\).

Step4: Factor the quadratic equation

Factor \(x^{2}-x - 56\). We need two numbers that multiply to \(-56\) and add up to \(-1\). The numbers are \(-8\) and \(7\). So, \(x^{2}-x - 56=(x - 8)(x+7)=0\).

Step5: Solve for \(x\)

Set each factor equal to zero: \(x - 8 = 0\) gives \(x = 8\); \(x+7 = 0\) gives \(x=-7\). But since \(x-4>0\) (lengths of sides of a triangle must be positive), we discard \(x=-7\).

Answer:

\(x = 8\)