Let's analyze the given problem step by step:

Part (a): Calculate the value of $x$

The triangle is a right triangle, with the hypotenuse being $AC$ and one leg as $AB = 4$. The horizontal leg $BC$ has a total length of $x + 5$, where the segment $AB$ starts from $x - 1$.

Using the Pythagorean theorem for the triangle: $AB^2 + BC^2 = AC^2$

1. $AB = 4$ (vertical side)
2. $BC = (x - 1) + (x + 5) = 2x + 4$
3. $AC = x + 5$

Substitute these values: $4^2 + (2x + 4)^2 = (x + 5)^2$

Simplify: $16 + (2x + 4)^2 = (x + 5)^2$

Expand both sides: $16 + 4x^2 + 16x + 16 = x^2 + 10x + 25$

Simplify further: $4x^2 + 16x + 32 = x^2 + 10x + 25$

Rearrange all terms: $4x^2 - x^2 + 16x - 10x + 32 - 25 = 0$

$3x^2 + 6x + 7 = 0$

Solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here:

• $a = 3$
• $b = 6$
• $c = 7$

Substitute: $x = \frac{-6 \pm \sqrt{6^2 - 4(3)(7)}}{2(3)}$

Simplify: $x = \frac{-6 \pm \sqrt{36 - 84}}{6}$

$x = \frac{-6 \pm \sqrt{-48}}{6}$

The discriminant is negative, which means there is no real solution for $x$. It appears there may be an inconsistency in the diagram or question setup.

Part (b): Calculate $E = \tan\theta + \sqrt{5} \cdot \sec\theta$

If we assume the triangle dimensions resolve to valid values of $\theta$, then:

1. Define trigonometric ratios:

   • $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{x - 1}$
   • $\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{x + 5}{x - 1}$

2. Substitute these into $E$: $E = \tan\theta + \sqrt{5} \cdot \sec\theta$

   $E = \frac{4}{x - 1} + \sqrt{5} \cdot \frac{x + 5}{x - 1}$

   Combine into a single fraction: $E = \frac{4 + \sqrt{5}(x + 5)}{x - 1}$

   Simplify further: $E = \frac{4 + \sqrt{5}x + 5\sqrt{5}}{x - 1}$

To calculate a numerical value for $E$, a valid $x$ is needed, which was not determined due to the inconsistency in part (a).

Would you like me to revisit part (a), explore alternative solutions, or clarify further? Let me know! Here are five follow-up questions for you:

1. Do you want me to recheck the equation derived in part (a)?
2. Should I explain how to handle the negative discriminant for $x$?
3. Would you like me to further elaborate on the trigonometric relationships in part (b)?
4. Do you want me to plot the functions to visualize their behavior?
5. Should I help simplify $E$ further for theoretical analysis?

Tip: Always verify the dimensions and relationships in diagrams carefully to ensure consistent problem-solving!