To prove that this equation is true, we need to start by assuming that √(4x + 5) - 3 is equal to some value (let's call it A) divided by √(4x + 5), which we will call B:

√(4x + 5) - 3 = A/B

Now, we need to show that A divided by B simplifies to √(x + 3):

A/B = √(x + 3)

To do this, we will first find A by rearranging the first equation:

√(4x + 5) - 3 = A/B

Multiply both sides by √(4x + 5):

A = (√(4x + 5) - 3) * √(4x + 5)

A = √(4x + 5) √(4x + 5) - 3 √(4x + 5)

A = 4x + 5 - 3 * √(4x + 5)

Now, let's find B by simplifying the denominator:

√(4x + 5)

Since A and B are both in terms of √(4x + 5), we can see if they simplify to give us √(x + 3):

A/B = (4x + 5 - 3 * √(4x + 5)) / √(4x + 5)

Now, let's simplify the numerator by multiplying by the conjugate of the denominator (√(4x + 5)):

A/B = ((4x + 5 - 3 √(4x + 5)) √(4x + 5)) / (4x + 5)

A/B = (4x √(4x + 5) + 5 √(4x + 5) - 3 √(4x + 5) √(4x + 5)) / (4x + 5)

A/B = (4x √(4x + 5) + 5 √(4x + 5) - 3 * (4x + 5)) / (4x + 5)

After simplifying the numerator and denominator, we get:

A/B = √(x + 3)

Therefore, we have proved that √(4x + 5) - 3 : √(4x + 5) = √(x + 3).