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To solve the problem of finding the remainder when \( 4x^2 - 6x + 7 \) is divided by \( x - 2 \), we can use the **Remainder Theorem**. 

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is simply \( f(a) \).

### Steps:
1. The polynomial in the problem is \( f(x) = 4x^2 - 6x + 7 \).
2. The divisor is \( x - 2 \), so we take \( a = 2 \).
3. To find the remainder, substitute \( x = 2 \) into the polynomial \( f(x) \):

\[
f(2) = 4(2)^2 - 6(2) + 7
\]
\[
f(2) = 4(4) - 12 + 7
\]
\[
f(2) = 16 - 12 + 7 = 11
\]

Thus, the remainder is **11**.

Would you like more details on this, or have any further questions?

Here are five related questions to expand on this topic:
1. How can we apply synthetic division to verify the remainder?
2. What is the full quotient when dividing \( 4x^2 - 6x + 7 \) by \( x - 2 \)?
3. How does the Remainder Theorem help in finding remainders for higher-degree polynomials?
4. Can this technique be used if the divisor is \( x + a \) instead of \( x - a \)?
5. What is the general form of the remainder when dividing a polynomial by a linear divisor?

**Tip**: The Remainder Theorem is a quick way to find remainders, but synthetic division provides more insight into the full quotient structure.

The remainder when 4x^2 - 6x + 7 is divided by x - 2 is ?

Learn how to find the remainder when dividing the quadratic polynomial 4x^2 - 6x + 7 by x - 2 using the Remainder Theorem. This step-by-step guide explains key algebra concepts and is suitable for students in Grades 8-10.

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