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The task requires finding the standard inner product of two polynomials \( p(x) \) and \( q(x) \) on the space \( P_2 \), the space of polynomials of degree at most 2.

The **standard inner product** of two polynomials \( p(x) = a_0 + a_1 x + a_2 x^2 \) and \( q(x) = b_0 + b_1 x + b_2 x^2 \) is defined as:

\[
\langle p, q \rangle = a_0 b_0 + a_1 b_1 + a_2 b_2
\]

### Problem 11:
Polynomials are:
\[
p(x) = -2 + x + 3x^2
\]
\[
q(x) = 4 - 7x^2
\]

We identify the coefficients of each polynomial:
- \( p(x) \) has coefficients \( a_0 = -2 \), \( a_1 = 1 \), \( a_2 = 3 \).
- \( q(x) \) has coefficients \( b_0 = 4 \), \( b_1 = 0 \), \( b_2 = -7 \).

Now, calculate the inner product:
\[
\langle p, q \rangle = (-2)(4) + (1)(0) + (3)(-7) = -8 + 0 - 21 = -29
\]

So, the inner product for Problem 11 is:
\[
\boxed{-29}
\]

### Problem 12:
Polynomials are:
\[
p(x) = -5 + 2x + x^2
\]
\[
q(x) = 3 + 2x - 4x^2
\]

We identify the coefficients of each polynomial:
- \( p(x) \) has coefficients \( a_0 = -5 \), \( a_1 = 2 \), \( a_2 = 1 \).
- \( q(x) \) has coefficients \( b_0 = 3 \), \( b_1 = 2 \), \( b_2 = -4 \).

Now, calculate the inner product:
\[
\langle p, q \rangle = (-5)(3) + (2)(2) + (1)(-4) = -15 + 4 - 4 = -15
\]

So, the inner product for Problem 12 is:
\[
\boxed{-15}
\]

### Summary:
- The inner product for Problem 11 is \( -29 \).
- The inner product for Problem 12 is \( -15 \).

Would you like any further clarifications or a deeper explanation on these calculations?

### Related Questions:
1. How does the inner product in \( P_2 \) generalize for polynomials of higher degrees?
2. Can the standard inner product in \( P_2 \) be applied to non-polynomial functions?
3. What are the properties of inner product spaces in linear algebra?
4. How does orthogonality work in polynomial spaces with the standard inner product?
5. How does the inner product differ from polynomial multiplication?

### Tip:
When working with polynomials and inner products, ensure you match the corresponding coefficients of like terms correctly to avoid calculation errors.

Find the standard inner product on P2 of the given polynomials:
11. p(x) = -2 + x + 3x^2, q(x) = 4 - 7x^2
12. p(x) = -5 + 2x + x^2, q(x) = 3 + 2x - 4x^2

Explore detailed solutions for finding the standard inner product of polynomials in P2. This includes problems with specific polynomials such as p(x) = -2 + x + 3x^2 and q(x) = 4 - 7x^2. Suitable for undergraduate students learning inner product spaces in linear algebra.

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