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Step-by-step explanation:

To find the value of b² + c², we can use the given equation and try to manipulate it to obtain the desired expression.

Given: b³ + c³ = 3bc - 1

Let's try to express b² + c² in terms of b³ + c³.

Starting with the given equation, we can rewrite it as:

(b³ + c³) + 1 = 3bc

Now, let's try to factor the left-hand side:

(b + c)(b² - bc + c²) + 1 = 3bc

Now, we have an expression involving b + c and b² - bc + c². We want to find b² + c², so let's focus on that part.

We can rearrange the equation to isolate b² + c²:

(b + c)(b² - bc + c²) + 1 = 3bc

(b² - bc + c²) = (3bc - 1) / (b + c)

Now, we can substitute the expression for b² - bc + c² back into the equation:

b² + c² = (3bc - 1) / (b + c)

Therefore, b² + c² is equal to (3bc - 1) divided by (b + c).

Please note that this is the simplified expression for b² + c² in terms of the given equation.