(x+4)(x+3)=2
To solve the equation \((x + 4)(x + 3) = 2\), we can first expand the left-hand side:
\[ (x + 4)(x + 3) = x^2 + 3x + 4x + 12 = x^2 + 7x + 12 \]
Now, set the equation equal to 2:
\[ x^2 + 7x + 12 = 2 \]
Next, subtract 2 from both sides to set the equation to zero:
\[ x^2 + 7x + 12 - 2 = 0 \]
This simplifies to:
\[ x^2 + 7x + 10 = 0 \]
Now, we need to factor the quadratic equation. We look for two numbers that multiply to 10 and add to 7. The numbers 5 and 2 satisfy this:
\[ (x + 5)(x + 2) = 0 \]
Now, set each factor equal to zero:
- \(x + 5 = 0 \quad \Rightarrow \quad x = -5\)
- \(x + 2 = 0 \quad \Rightarrow \quad x = -2\)
Thus, the solutions to the equation \((x + 4)(x + 3) = 2\) are:
\[ x = -5 \quad \text{and} \quad x = -2 \]
You can verify these solutions by substituting them back into the original equation:
For \(x = -5\): \[ (-5 + 4)(-5 + 3) = (-1)(-2) = 2 \]
For \(x = -2\): \[ (-2 + 4)(-2 + 3) = (2)(1) = 2 \]
Both solutions are correct. Therefore, the final solutions are:
\[ \boxed{-5 \text{ and } -2} \]
