To solve the equation, we need to isolate the variable x. Let's simplify the equation step by step:

Starting equation: căn(4x + 20) + căn(x + 5) - 1/3 căn(9x + 45) = 4

First, let's simplify the square roots:

√(4x + 20) + √(x + 5) - 1/3 √(9x + 45) = 4

Now, let's isolate the square roots on one side of the equation:

√(4x + 20) + √(x + 5) = 4 + 1/3 √(9x + 45)

Next, let's square both sides of the equation to eliminate the square roots:

(√(4x + 20) + √(x + 5))^2 = (4 + 1/3 √(9x + 45))^2

Expanding both sides:

(4x + 20) + 2√((4x + 20)(x + 5)) + (x + 5) = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

Now, let's simplify:

4x + 20 + 2√(4x^2 + 20x + 20x + 100) + x + 5 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

4x + 20 + 2√(4x^2 + 40x + 100) + x + 5 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

4x + 20 + 2√((2x + 10)^2) + x + 5 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

4x + 20 + 2(2x + 10) + x + 5 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

4x + 20 + 4x + 20 + x + 5 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

9x + 45 + 5x + 25 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

14x + 70 = 16 + 8/3 √(9x + 45) + 1/9 (9x + 45)

Now, let's isolate the square root term:

14x + 70 - 16 - 8/3 √(9x + 45) - 1/9 (9x + 45) = 0

14x + 54 - 8/3 √(9x + 45) - 1/9 (9x + 45) = 0

Now, let's simplify the fractions:

14x + 54 - 24/3 √(9x + 45) - 1/9 (9x + 45) = 0

14x + 54 - 8 √(9x + 45) - 1/9 (9x + 45) = 0

Now, let's combine like terms:

14x - 8 √(9x + 45) + 54 - 1/9 (9x + 45) = 0

Now, let's simplify the expression:

14x - 8 √(9x + 45) + 54 - (1/9) (9x + 45) = 0

14x - 8 √(9x + 45) + 54 - x - 5 = 0

13x - 8 √(9x + 45) + 49 = 0

Now, let's isolate the square root term:

-8 √(9x + 45) = -13x - 49

Now, let's square both sides to eliminate the square root:

(-8 √(9x + 45))^2 = (-13x - 49)^2

64(9x + 45) = 169x^2 + 338x + 2401

576x + 2880 = 169x^2 + 338x + 2401

Rearranging the equation:

169x^2 + 338x - 576x - 2401 + 2880 = 0

169x^2 - 238x + 479 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.