1. We are asked to express the quadratic expression $19 - 4x + x^2$ in the form $(x + a)^2 + b$, where $a$ and $b$ are integers. 2. The general approach is to complete the square. Recall that: $$ (x + a)^2 = x^2 + 2ax + a^2 $$ 3. Start with the given expression: $$ 19 - 4x + x^2 $$ 4. Rewrite it in standard quadratic form: $$ x^2 - 4x + 19 $$ 5. To complete the square, focus on the $x^2 - 4x$ part. Take half of the coefficient of $x$, which is $-4$, so half is $-2$. Square it: $$ (-2)^2 = 4 $$ 6. Add and subtract this square inside the expression to keep it equivalent: $$ x^2 - 4x + 4 - 4 + 19 $$ 7. Group the perfect square trinomial and simplify the constants: $$ (x^2 - 4x + 4) + (19 - 4) = (x - 2)^2 + 15 $$ 8. Notice that $(x - 2)^2$ can be written as $(x + a)^2$ with $a = -2$. 9. Therefore, the expression in the desired form is: $$ (x - 2)^2 + 15 = (x + (-2))^2 + 15 $$ 10. So the values are: $$ a = -2, \quad b = 15 $$