To determine whether [tex]\((x+6)(x+5) = x^2 + 11x + 30\)[/tex], we need to expand the left-hand side of the equation and then compare it to the right-hand side. Let's go through this step-by-step.
1. Expand the left side of the equation:
[tex]\[
(x+6)(x+5)
\][/tex]
Using the distributive property (also known as the FOIL method for binomials), we get:
[tex]\[
(x+6)(x+5) = x(x+5) + 6(x+5)
\][/tex]
Now, distribute [tex]\(x\)[/tex] and [tex]\(6\)[/tex] to the terms inside the parentheses:
[tex]\[
= x \cdot x + x \cdot 5 + 6 \cdot x + 6 \cdot 5
\][/tex]
[tex]\[
= x^2 + 5x + 6x + 30
\][/tex]
Next, combine the like terms:
[tex]\[
= x^2 + (5x + 6x) + 30
\][/tex]
[tex]\[
= x^2 + 11x + 30
\][/tex]
2. Compare the expanded form with the right side of the equation:
The expanded form [tex]\(x^2 + 11x + 30\)[/tex] matches exactly with the right side of the equation [tex]\(x^2 + 11x + 30\)[/tex].
3. Conclusion:
Since both sides of the equation are the same, we can conclude that:
[tex]\[
(x+6)(x+5) = x^2 + 11x + 30
\][/tex]
The equation holds true, so the answer to the question is yes.
