To solve the equation $(2x + 3)^2 + (x + 4)^2 = 10$, follow these steps:

Step 1: Expand both squares

$(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$ $(x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$

Step 2: Add the expanded terms

$(4x^2 + 12x + 9) + (x^2 + 8x + 16) = 10$ $4x^2 + x^2 + 12x + 8x + 9 + 16 = 10$ $5x^2 + 20x + 25 = 10$

Step 3: Simplify the equation

Subtract 10 from both sides: $5x^2 + 20x + 25 - 10 = 0$ $5x^2 + 20x + 15 = 0$

Step 4: Divide the equation by 5

$x^2 + 4x + 3 = 0$

Step 5: Solve the quadratic equation

Factor the quadratic equation: $(x + 1)(x + 3) = 0$

Step 6: Find the solutions

Set each factor equal to zero: $x + 1 = 0 \quad \text{or} \quad x + 3 = 0$ $x = -1 \quad \text{or} \quad x = -3$

Thus, the solutions are $x = -1$ and $x = -3$.

Would you like more details or have any questions?

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Here are 5 related questions for further practice:

1. Solve $(3x + 2)^2 + (x - 1)^2 = 25$.
2. Solve the quadratic equation $2x^2 + 7x + 3 = 0$.
3. Factor the quadratic expression $x^2 + 5x + 6 = 0$.
4. Find the roots of $x^2 + 2x - 8 = 0$.
5. Solve $(x + 5)^2 + (2x + 1)^2 = 13$.

Tip: When solving quadratic equations, always check if they can be factored before resorting to the quadratic formula. It saves time!