To factorize the expression [tex]\( x^2 + 7x + 10 \)[/tex], we need to find two numbers that multiply to 10 and add up to 7.
The numbers that satisfy this condition are 2 and 5.
So, we can write the expression as the product of two binomials:
[tex]\[ x^2 + 7x + 10 = (x + 2)(x + 5) \][/tex]
Now, let's use this factorization to solve the equation [tex]\( x^2 + 7x + 10 = 0 \)[/tex].
Setting each factor equal to zero:
[tex]\[ (x + 2) = 0 \quad \text{or} \quad (x + 5) = 0 \][/tex]
Solving these equations:
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
[tex]\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \][/tex]
So, the solutions to the equation [tex]\( x^2 + 7x + 10 = 0 \)[/tex] are:
[tex]\[ x = -2 \quad \text{and} \quad x = -5 \][/tex]
