Find the Coefficients: Solving (ax-1)² + (bx-2)² = 5x²-8x+5

Fill in the blanks:

$(?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5$

To solve this problem, we'll fill in the blanks in the expression $(?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5$.

Let's expand the individual expressions:
1. $(a \cdot x - 1)^2 = a^2x^2 - 2ax + 1$
2. $(b \cdot x - 2)^2 = b^2x^2 - 4bx + 4$

Now, add these expansions together:
$(a^2x^2 - 2ax + 1) + (b^2x^2 - 4bx + 4)$
= $(a^2 + b^2)x^2 - (2a + 4b)x + 5$

We equate this to the given quadratic: $5x^2 - 8x + 5$.

Matching coefficients, we have:
$a^2 + b^2 = 5$
$2a + 4b = 8$

Now, solve these equations:

From the second equation: $2a + 4b = 8$, divide by 2:
$a + 2b = 4$

Substitute $a = 4 - 2b$ from this into $a^2 + b^2 = 5$:

$(4 - 2b)^2 + b^2 = 5$
$16 - 16b + 4b^2 + b^2 = 5$
$5b^2 - 16b + 16 = 5$
$5b^2 - 16b + 11 = 0$

Factoring the quadratic equation $5b^2 - 16b + 11 = 0$:
$(5b - 11)(b - 1) = 0$

This gives us $b = \frac{11}{5}$ or $b = 1$.

Using $b = 1$, substitute back to find $a$:
$a = 4 - 2(1) = 2$

Thus, the correct integers for the blanks are $a = 2$ and $b = 1$.

Therefore, the correct answer is $2, \text{ } 1$.