To find the maximum value of the expression x^2 + 14x + 9 + 22x + 3, we first simplify it.
Combining like terms, we can rewrite the expression as:
x^2 + (14x + 22x) + (9 + 3) = x^2 + 36x + 12.
This expression is a quadratic equation in the standard form of ax^2 + bx + c where a = 1, b = 36, and c = 12.
Since the coefficient of x^2 (which is a) is positive, the parabola opens upwards. Therefore, this expression does not have a maximum value; it can go to infinity as x increases.
However, we can find the minimum value of the expression by using the formula for the vertex of a parabola, which occurs at x = -b/(2a). Here, substituting in the values gives:
x = -36/(2*1) = -18.
Substituting x = -18 back into the expression to find the minimum:
= (-18)^2 + 36*(-18) + 12 = 324 – 648 + 12 = -312.
Therefore, the minimum value of the expression is -312, and there is no maximum value since it can extend to positive infinity.