To solve the given equation [tex]\((x+4)(x+6) = x^2 + mx + k\)[/tex], we need to expand the left-hand side and then compare it with the right-hand side to identify the values of [tex]\(m\)[/tex] and [tex]\(k\)[/tex].
Let's start by expanding the left-hand side:
[tex]\[(x+4)(x+6)\][/tex]
We distribute each term in the first binomial across the second binomial:
[tex]\[= x \cdot x + x \cdot 6 + 4 \cdot x + 4 \cdot 6\][/tex]
[tex]\[= x^2 + 6x + 4x + 24\][/tex]
Combining like terms, we get:
[tex]\[= x^2 + (6x + 4x) + 24\][/tex]
[tex]\[= x^2 + 10x + 24\][/tex]
Now, we compare this expanded form with the given equation [tex]\(x^2 + mx + k\)[/tex]:
[tex]\[x^2 + 10x + 24 = x^2 + mx + k\][/tex]
By comparing the coefficients of [tex]\(x\)[/tex] and the constant terms, we see that:
[tex]\[m = 10\][/tex]
[tex]\[k = 24\][/tex]
Now, we need to find the value of [tex]\(m + k\)[/tex]:
[tex]\[m + k = 10 + 24 = 34\][/tex]
Hence, the value of [tex]\(m + k\)[/tex] is:
[tex]\[\boxed{34}\][/tex]
